ALBERT

Description: Entrainment of xenoliths and their consequent assimilation are key processes in modifying the crystallization kinetics and magma dynamics of conduit systems. Here, an integrated textural and mineral chemical study of the evolution of a suite of gabbroic inclusions within a set of sheet intrusions from the Ardnamurchan Central Complex, NW Scotland, is presented. The key findings are as follows: (1) the host magma sheets and inclusions are not cognate; (2) there are microstructural and mineral chemical similarities between the gabbroic inclusions and the textures and mineralogy of the major Hypersthene Gabbro on Ardnamurchan; (3) orientations of magnetic fabrics within the host sheet groundmass and within the gabbroic inclusions are virtually identical. Field observations suggest that the inclusions were derived from the Hypersthene Gabbro and were entrained in a few laterally restricted magma segments that subsequently coalesced with inclusion-free segments into continuous sheets. Using Stokes’ Law and adaptations thereof, we calculate that the magma within the inclusion-free segments behaved as a Newtonian fluid, with a potential settling velocity of 〈0·028 m s –1 . In contrast, the presence of gabbro inclusions probably modified the magma dynamics to Bingham-like behaviour. We infer that this variation in the magma rheology of separate segments continued after coalescence and internally partitioned the magma sheet, preventing lateral mixing and inclusion transport.

Description: The Varberg–Torpa charnockite–granite association (Varberg, SW Sweden) consists of the magmatic Varberg charnockite (1399 ± 6 Ma) and the Torpa granite (1380 ± 12 Ma). The Torpa granite is both continuous and, based on its whole-rock geochemistry, synmagmatic with the Varberg charnockite. The granite body also contains a number of charnockite inliers. P – T estimation using garnet–clinopyroxene and orthopyroxene–clinopyroxene Fe–Mg exchange thermometry and garnet–orthopyroxene–plagioclase–quartz barometry gives temperatures and pressures (750–850°C; 800–850 MPa) that most probably approximate the P – T conditions during emplacement of the charnockite compared with a lower crystallization temperature (650–700°C) for the granite. The earliest recognized fluid inclusions in both the granite and charnockite consist of H 2 O–CO 2 mixtures (H 2 O volume fraction 0·2–0·7). Fluid inclusions in the charnockite are characterized by high CO 2 densities (up to 1·0 g cm – 3 ; 40–90% bulk CO 2 ), of probable magmatic origin, and are best preserved in garnet, plagioclase, and fluorapatite (in order of decreasing CO 2 densities), and sometimes also in clinopyroxene. Fluid inclusions with the highest CO 2 densities (1·08–1·10 g cm – 3 ) are found in quartz ( T h –31 to –36°C) and may have originated under high P – T conditions during emplacement and cooling of the charnockite. Magmatic fluids in the granite correspond to aqueous–carbonic inclusions with an estimated bulk composition (mol %) of H 2 O 73%, CO 2 25%, NaCl 2%. The salinity of the solutes in the granite (typically 14–20 wt % NaCl-eq.) is generally higher than for the charnockite (0–8 wt % NaCl-eq.). Field, petrographic, mineralogical, geochemical, and fluid inclusion evidence indicates that, compared with the H 2 O-rich granite, the magma responsible for the charnockite had a preponderance of CO 2 over H 2 O, which lowered the H 2 O activity in the melt, stabilizing ortho- and clinopyroxene. This evidence also supports the idea that the granite and charnockite were derived from a common source magma (most probably a fluid-rich basalt at the base of the crust) as a result of fractional crystallization.

Description: We performed crystallization experiments at 2–3 GPa at 700–950°C on basaltic and pelitic lithologies with added water and sulphur to constrain the factors controlling sulphur behaviour in subduction zones and how it may have varied through geological time. The resulting hydrous silicic melts have up to 20 times more dissolved sulphur (up to 1 wt %) than at 0·2–0·4 GPa, when moderately oxidized conditions prevail. Such high solubilities appear to result from the combined effects of enhanced solubility of water in high-pressure silicate melts (10–20 wt % H 2 O), which acts to decrease silica activity, and oxidizing conditions. The results confirm previous findings that high sulphur contents in silicate melts do not necessarily require iron-rich compositions, suggesting instead that sulphur–water complexes play a fundamental role in sulphur dissolution mechanisms in iron-poor silicic melts, in agreement with recent spectroscopic data. The experimental melts reproduce Phanerozoic slab-derived magmas, in particular their distinct Ca- and Mg-rich composition. The results also show that sulphur increases the degree of melting of basalt lithologies. Hence, we suggest that subducted slabs will preferentially melt where sulphur is present in abundance and that the variability in arc magma sulphur output reflects, in part, the vagaries of sulphur distribution in the slab source. In contrast, comparison with the composition of Archean felsic rocks suggests that, in the early Earth, much less sulphur was present in subducted slabs, in agreement with a number of independent lines of evidence showing that the Archean ocean, hence the hydrothermally altered subducted Archean oceanic crust, was considerably poorer in sulphur than at present. Volcanic degassing of sulphur was thus probably much weaker during the Archean than in Proterozoic–Phanerozoic times.

Description: Anak Krakatau is a basaltic andesite cone that has grown following the famous caldera-forming 1883 eruption of Krakatau. It breached sea level in 1927 and since the 1950s has been growing at an average rate of ~8 cm a week. We present new major and trace element data combined with whole-rock 18 O, Sr and Nd isotope data for 1883, 1993 and 2002 Krakatau eruptive products and the surrounding crust. Bombs erupted from Anak Krakatau during 2002 contain frothy metasedimentary and plutonic xenoliths that show variable degrees of thermal metamorphism, plastic deformation and partial melting. Contact-metamorphic minerals such as cordierite and tridymite in metasedimentary xenoliths are consistent with high-temperature metamorphism and incorporation at mid- to upper-crustal depth. Energy-constrained assimilation and fractional crystallization modelling of whole-rock data suggests that the Anak Krakatau magmas have a genetic relationship with the 1883 eruption products. The geochemical impact of crustal contaminants on whole-rock compositions is apparently small, and we conclude that low levels of assimilation of a quartzo-feldspathic sediment are recorded in Anak Krakatau magmas. Plagioclase phenocrysts from the 2002 eruption exhibit disequilibrium textures and complex compositional zoning, however, and are also isotopically variable with a total range in 87 Sr/ 86 Sr of 0·7043–0·7048 as determined by in situ laser ablation inductively coupled plasma mass spectrometry. This suggests that although shallow crustal assimilation appears to have had a limited effect on whole-rock chemistry, a complex late-stage differentiation history is recorded within the magma’s cargo of crystals and xenoliths.

Description: The celebrated truncated T-matrix method for wave propagation models belongs to a class of the reduced basis methods (RBMs), with the parameters being incident waves and incident directions. The T-matrix characterizes the scattering properties of the obstacles independent of the incident and receiver directions. In the T-matrix method the reduced set of basis functions for representation of the scattered field is constructed analytically and hence, unlike other classes of the RBM, the T-matrix RBM avoids computationally intensive empirical construction of a reduced set of parameters and the associated basis set. However, establishing a convergence analysis and providing practical a priori estimates for reducing the number of basis functions in the T-matrix method has remained an open problem for several decades. In this work we solve this open problem for time-harmonic acoustic scattering in two and three dimensions. We numerically demonstrate the convergence analysis and the a priori parameter estimates for both point-source and plane-wave incident waves. Our approach can be used in conjunction with any numerical method for solving the forward wave propagation problem.

Description: We derive an algorithm for the adaptive approximation of solutions to parabolic equations. It is based on adaptive finite elements in space and the implicit Euler discretization in time with adaptive time-step sizes. We prove that, given a positive tolerance for the error, the adaptive algorithm reaches the final time with a space–time error between continuous and discrete solution that is below the given tolerance. Numerical experiments reveal a more than competitive performance of our algorithm ASTFEM (adaptive space–time finite element method).

Description: We propose and analyse a new family of nonconforming elements for the Brinkman problem of porous media flow. The corresponding finite element methods are robust with respect to the limiting case of Darcy flow, and the discretely divergence-free functions are in fact divergence-free. Therefore, in the absence of sources and sinks, the method is strongly mass-conservative. We also show how the proposed elements are part of a discrete de Rham complex.

Description: In this paper we investigate the superconvergence of local discontinuous Galerkin (LDG) methods for solving one-dimensional linear time-dependent fourth-order problems. We prove that the error between the LDG solution and a particular projection of the exact solution, e u , achieves th-order superconvergence when polynomials of degree k ( k ≥ 1) are used. Numerical experiments with P k polynomials, with 1 ≤ k ≤ 3, are displayed to demonstrate the theoretical results, which show that the error e u actually achieves ( k +2)th-order superconvergence, indicating that the error bound for e u obtained in this paper is suboptimal. Initial boundary value problems, nonlinear equations and solutions having singularities, are numerically investigated to verify that the conclusions hold true for very general cases.

Description: The discrete mollification method, a convolution-based filtering procedure for the regularization of illposed problems, is applied here to stabilize explicit schemes, which were first analysed by Karlsen & Risebro (2001, An operator splitting method for nonlinear convection–diffusion equations. M2AN Math. Model. Numer. Anal. 35 , 239–269) for the solution of initial value problems of strongly degenerate parabolic partial differential equations in two space dimensions. Two new schemes are proposed, which are based on directionwise and two-dimensional discrete mollification of the second partial derivatives forming the Laplacian of the diffusion function. The mollified schemes permit substantially larger time steps than the original (basic) scheme. It is proven that both schemes converge to the unique entropy solution of the initial value problem. Numerical examples demonstrate that the mollified schemes are competitive in efficiency, and in many cases significantly more efficient, than the basic scheme.

Description: A family of explicit adaptive algorithms is designed to solve nonlinear scalar one-dimensional conservation laws. Based on the Godunov scheme on a uniform grid, a first strategy uses the multiresolution analysis of the solution to design an adaptive grid that evolves in time according to the time-dependent local smoothness. The method is furthermore enhanced by a local time-stepping strategy. Both numerical schemes are shown to converge towards the unique entropy solution.

Description: In this paper the first error analyses of hybridizable discontinuous Galerkin (HDG) methods for convection–diffusion equations for variable-degree approximations and nonconforming meshes are presented. The analysis technique is an extension of the projection-based approach recently used to analyse the HDG method for the purely diffusive case. In particular, for approximations of degree k on all elements and conforming meshes, we show that the order of convergence of the error in the diffusive flux is k + 1 and that of a projection of the error in the scalar unknown is 1 for k = 0 and k + 2 for k 〉 0. When nonconforming meshes are used our estimates do not rule out a degradation of 1/2 in the order of convergence in the diffusive flux and a loss of 1 in the order of convergence of the projection of the error in the scalar variable. However, they do guarantee the optimal convergence of order k + 1 of the scalar variable.

Description: Fast multigrid solvers are considered for the linear systems arising from the bilinear finite element discretizations of second-order elliptic equations with anisotropic diffusion. Optimal convergence of Vcycle multigrid methods in the semicoarsening case and nearly optimal convergence of V-cycle multigrid method with line smoothing in the uniformly-coarsening case are established using the Xu-Zikatanov identity. Since the ‘regularity assumption’ is not used in the analysis, the results can be extended to general domains consisting of rectangles.

Description: In this paper we consider a class of incompressible viscous fluids whose viscosity depends on the shear rate and pressure. We deal with isothermal steady flow and analyse the Galerkin discretization of the corresponding equations. We discuss the existence and uniqueness of discrete solutions and their convergence to the solution of the original problem. In particular, we derive a priori error estimates, which provide optimal rates of convergence with respect to the expected regularity of the solution. Finally, we demonstrate the achieved results by numerical experiments. The fluid models under consideration appear in many practical problems, for instance, in elastohydrodynamic lubrication where very high pressures occur. Here we consider shear-thinning fluid models similar to the power-law/Carreau model. A restricted sublinear dependence of the viscosity on the pressure is allowed. The mathematical theory concerned with the self-consistency of the governing equations has emerged only recently. We adopt the established theory in the context of discrete approximations. To our knowledge, this is the first analysis of the finite element method for fluids with pressure-dependent viscosity. The derived estimates coincide with the optimal error estimates established recently for Carreau-type models, which are covered as a special case.

Description: Discrete maximum principles (DMPs) are established for finite element approximations of systems of nonlinear parabolic partial differential equations with mixed boundary and interface conditions. The results are based on an algebraic DMP for suitable systems of ordinary differential equations.

Description: The adaptive cubic regularization algorithm described in Cartis et al. (2009, Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results. Math. Program. , 127 , 245–295; 2010, Adaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function- and derivative-evaluation complexity [online]. Math. Program. , DOI: 10.1007/s10107-009-0337-y) is adapted to the problem of minimizing a nonlinear, possibly nonconvex, smooth objective function over a convex domain. Convergence to first-order critical points is shown under standard assumptions, without any Lipschitz continuity requirement on the objective's Hessian. A worst-case complexity analysis in terms of evaluations of the problem's function and derivatives is also presented for the Lipschitz continuous case and for a variant of the resulting algorithm. This analysis extends the best-known bound for general unconstrained problems to nonlinear problems with convex constraints.

Description: This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximation which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators in three dimensions (3D). Following the so-called DDFV (discrete duality finite volume) approach developed by Hermeline (1998, Une méthode de volumes finis pour les équations elliptiques du second ordre. C. R. Math. Acad. Sci. Paris , 326 , 1433–1436 (in French); 2000, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. , 160 , 481–499) and by Domelevo & Omnès (2005, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. M2AN Math. Model. Numer. Anal. , 39 , 1203–1249) in 3D, we consider a ‘double’ covering T of a 3D domain by a rather general primal mesh and by a well-chosen ‘dual’ mesh. The associated discrete divergence operator div  T is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator T is defined by a local affine interpolation that takes into account the geometry of the double mesh. Under mild geometrical constraints on the choice of the dual volumes, we show that –div  T and T are linked by the ‘discrete duality property’, which is an analogue of the integration-by-parts formula. The primal mesh need not be conformal, and its interfaces can be general polygons. We give several numerical examples for anisotropic linear diffusion problems; good convergence properties are observed. The sequel, Andreianov et al. (2011a, On 3D DDFV discretization of gradient and divergence operators. II. Discrete functional analysis tools and applications to degenerate parabolic problems. HAL preprint available at http://hal.archives-ouvertes.fr/hal-00567342 ) to this paper will summarize some key discrete functional analysis tools for DDFV schemes and give applications to proving convergence of DDFV schemes for several nonlinear degenerate parabolic partial differential equations.

Description: This work is concerned with the numerical implementation of the discrepancy principle for nonsmooth Tikhonov regularization for linear inverse problems. First, some theoretical properties of the solutions to the discrepancy equation, i.e., uniqueness and upper bounds, are discussed. Then, the idea of Padé approximation is exploited for designing model functions with model parameters iteratively updated. Two algorithms are proposed for its efficient numerical realization, i.e., a two-parameter algorithm based on model functions and a quasi-Newton method, and their convergence properties are briefly discussed. Numerical results for four nonsmooth models are presented to demonstrate the accuracy of the principle and to illustrate the efficiency and robustness of the proposed algorithms.

Description: This paper is devoted to the convergence analysis of the upwind finite volume scheme for the initialand boundary-value problems associated with the linear transport equation in any dimension, on general unstructured meshes. We are particularly interested in the case where the initial and boundary data are in L and the advection vector field has low regularity properties, namely L 1 (]0, T [, ( W 1,1 ()) d ), with suitable assumptions on its divergence. In this general framework, we prove uniform in time strong convergence in L p (), with p 〈 +, of the approximate solution towards the unique weak solution of the problem as well as the strong convergence of its trace. The proof relies, in particular, on the Friedrichs' commutator argument, which is classical in the renormalized solutions theory. Note that this result remains valid if the data are suitably approximated in L 1 . This is nothing but the discrete counterpart of the nice compactness properties deduced from the renormalized solution theory. We conclude with some numerical experiments showing that the convergence rate seems to be 1/2, as in the case of smoother advection fields, but this is still an open question up to now.

Description: In this paper, we study the existence, uniqueness and regularity properties of solutions for the nonstandard Volterra integral equation . We then present a collocation method to solve this equation, and analyse the convergence and superconvergence of piecewise polynomial collocation approximations. We also illustrate the theoretical results by extensive numerical experiments.

Description: In this paper we consider the discretization error in space and time of an H 1 gradient flow for an energy integral where the energy density is given by the sum of a double-well potential term and a bending energy term. We show that the problem is equivalent to a nonlinear heat equation with nonlocal nonlinearity and adapt the standard error analysis theory developed for the nonlinear heat equation to our case. In doing so we bound the discretization error in terms of the mesh size and time step as well as energy parameters. In particular, we carefully track how the size of the bending energy affects the error bounds.

Description: A local convergence analysis of Newton's method for finding a singularity of a differentiable vector field defined on a complete Riemannian manifold, based on the majorant principle, is presented in this paper. This analysis provides a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the vector field under consideration. It also allows us to obtain the optimal convergence radius and the biggest range for the uniqueness of the solution and to unify some previously unrelated results.

Description: For analytic functions we study the kernel of the remainder terms of Gaussian quadrature rules with respect to Bernstein–Szego weight functions where 0〈 α 〈 β , β != 2 α , | |〈 β – α , and whose denominator is an arbitrary polynomial of exact degree 2 that remains positive on [–1, 1]. The subcase α =1, β =2/(1+ ), –1〈 〈0 and =0 has been considered recently by Spalevíc, M. M. & Praníc, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234 , 1049–1057). The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds for the corresponding Gaussian quadratures. The approach we use in this paper, which is different from the one adopted in Spalevíc, M. M. & Praníc, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234 , 1049–1057), ensures that the actual conditions for determining the locations on the elliptic contours where the modulus of the kernel attains its maximum value are approximated very precisely.

Description: Phase equilibrium experiments were performed to determine the pre-eruptive conditions of the explosive eruption of Montaña Blanca (2020 bp ) that occurred from a satellite vent located on the east flank of Teide volcano (Tenerife). Crystallization experiments used a phonolitic obsidian from the fall-out deposit as the starting material; this contains 5 wt % anorthoclase, diopside and magnetite with minor amounts of biotite and ilmenite, set in a glassy matrix that contains microlites of Ca-rich alkali feldspar. Temperature was varied between 850 and 800°C, and pressure between 200 and 50 MPa. The oxygen fugacity ( f O 2 ) was varied between NNO + 0·2 (0·2 log units above the Ni–NiO solid buffer) and NNO – 2, and dissolved water contents varied from 7 to 1·5 wt %. Comparison between natural and experimental phase proportions and compositions indicates that the main body of phonolitic magma was stored at 850 ± 15°C, 50 ± 20 MPa, 2·5 ± 0·5 wt % H 2 O at an f O 2 around NNO – 0·5 prior to eruption, equivalent to depths of between 1 and 2 km below the surface. Some clinopyroxene crystals hosting H 2 O-rich melt inclusions possibly originate from an intermittent supply of phonolitic magma stored at somewhat deeper levels (100 MPa). The Ca- and Fe-rich composition of alkali feldspar phenocryst rims and microlites attests to the intrusion of a more mafic magma into the reservoir just prior to eruption; this is evidenced by the appearance of banded pumices in the later products of the eruptive sequence. The comparison with other phonolitic magmas from Tenerife and elsewhere (e.g. Vesuvius, Laacher See) shows that differences in the eruption dynamics of phonolitic magmas can be correlated with differences in magma storage depths, along with variations in pre-eruptive volatile contents.

Description: We examine the partial survival of high-temperature mantle microstructures throughout multi-stage hydration- and dehydration-mediated pseudomorphism at differing pressure–temperature–fluid conditions. Throughout the harzburgitic mantle section of the Leka Ophiolite Complex (Norway), finite domains of parallel olivine encompassed by mesh-textured olivine resembling ‘perfectly cleaved’ olivine grains were identified. Crystallographic orientation mapping, combined with micro-computed tomography, reveals that the parallel olivine grains are highly misoriented (up to 90°) with no crystal-preferred orientation, despite remaining parallel in three dimensions. Parallel olivine grains exhibit free dislocations with low dislocation density, whereas within mesh-textured olivine dislocations are aligned into walls. MnO is enriched (up to 1·8 wt %) and NiO depleted (0·21 ± 0·24 wt %) within parallel olivine grains compared with mesh-textured olivine (0·29 ± 0·14 wt % MnO; 0·38 ± 0·19 wt % NiO). Clinopyroxene lamellae that are crystal-plastically deformed occur sandwiched in lizardite layers between every parallel olivine grain or fully enclosed within olivine. Al 2 O 3 and Cr 2 O 3 concentrations of clinopyroxene lamellae (2·09 ± 0·88 wt % Al 2 O 3 ; 0·79 ± 0·27 wt % Cr 2 O 3 ) overlap with those of primary clinopyroxene grains (2·43 ± 0·69 wt % Al 2 O 3 ; 0·83 ± 0·36 wt % Cr 2 O 3 ) and are distinctly different from those of secondary diopside found within the parallel olivine domains. Intragranular serpentine inclusions (X Mg = 0·95 ± 0·01), displaying elevated Al 2 O 3 (3·92 ± 4·10 wt %) and Cr 2 O 3 (0·78 ± 0·82 wt %) concentrations, are exclusively found within parallel olivine grains. Lizardite (X Mg = 0·92 ± 0·02) within the domains originates from hydration of parallel olivine and compositionally overlaps with mesh-texture lizardite. Antigorite (X Mg = 0·95 ± 0·01) replaces both types of olivine grains. Whole-rock compositions indicate a harzburgitic composition; however, microstructural and chemical observations and the current absence of primary orthopyroxene suggest that the precursor silicate of every parallel olivine domain was a single orthopyroxene grain that was initially serpentinized and later dehydrated to result in the present microstructure. Although desilicification is necessary during the transformation of orthopyroxene to olivine via a bastite stage, calculations based on whole-rock compositions imply that the released SiO 2(aq) was mobile only over micrometer to centimeter scales, reacting with the surrounding olivine directly to form serpentine. Crosscutting relationships and serpentine compositions imply that dehydration occurred prior to the now evident lizardite- and antigorite-serpentinization. Comparison with the regional geological setting indicates that dehydration may have occurred transiently within the oceanic lithosphere prior to obduction.

Description: We observe void growth and coalescence into cavity-bearing shear bands during deformation of wet synthetic anorthite aggregates containing 〈3 vol. % silica-enriched melt. Samples were deformed in the Newtonian creep regime to high strain during torsion experiments at 1100°C and 400 MPa confining pressure. Localized cavity-bearing shear bands show an S–C'-geometry: the bands (C') are oriented at about 30° to the compression direction of the imposed simple shear and the internal foliation (S) of the bands is rotated towards the horizontal external shear plane. Cavity-bearing shear bands started to nucleate in the sample periphery above a shear strain threshold of 2. Quartz crystallized from the water-saturated SiO 2 -rich melt within large cavities inside these bands, which requires that the melt is decompressed by 〉200 MPa during their formation. The dynamically evolving cavities are sites of locally reduced pressure that collect the melt distributed in the adjacent matrix. Therefore, cavitation damage under ductile conditions may result in the development of an efficient melt channelling system controlling SiO 2 -rich melt flow in the lower crust. Electron backscatter diffraction analysis shows that the quartz inside the cavity bands has a crystallographic preferred orientation (CPO). The development of the CPO is explained by the preferred dissolution of crystals oriented with the rhombohedra and trigonal dipyramids orthogonal to the compression direction and by preferential growth of crystals aligned with their 〈0001〉 axis in the extension direction of the externally applied simple shear deformation.

Description: The phase relations have been investigated experimentally at 200 and 500 MPa as a function of water activity for one of the least evolved (Indian Batt Rhyolite) and of a more evolved rhyolite composition (Cougar Point Tuff XV) from the 12·8–8·1 Ma Bruneau–Jarbidge eruptive center of the Yellowstone hotspot. Particular priority was given to accurate determination of the water content of the quenched glasses using infrared spectroscopic techniques. Comparison of the composition of natural and experimentally synthesized phases confirms that high temperatures (〉900°C) and extremely low melt water contents (〈1·5 wt % H 2 O) are required to reproduce the natural mineral assemblages. In melts containing ~0·5–1·5 wt % H 2 O, the liquidus phase is clinopyroxene (excluding Fe–Ti oxides, which are strongly dependent on fO 2 ), and the liquidus temperature of the more evolved Cougar Point Tuff sample (BJR; ~940–1000°C) is at least 30°C lower than that of the Indian Batt Rhyolite lava sample (IBR2; 970–1030°C). For the composition BJR, the comparison of the compositions of the natural and experimental glasses indicates a pre-eruptive temperature of at least 900°C. The composition of clinopyroxene and pigeonite pairs can be reproduced only for water contents below 1·5 wt % H 2 O at 900°C, or lower water contents if the temperature is higher. For the composition IBR2, a minimum temperature of 920°C is necessary to reproduce the main phases at 200 and 500 MPa. At 200 MPa, the pre-eruptive water content of the melt is constrained in the range 0·7–1·3 wt % at 950°C and 0·3–1·0 wt % at 1000°C. At 500 MPa, the pre-eruptive temperatures are slightly higher (by ~30–50°C) for the same ranges of water concentration. The experimental results are used to explore possible proxies to constrain the depth of magma storage. The crystallization sequence of tectosilicates is strongly dependent on pressure between 200 and 500 MPa. In addition, the normative Qtz–Ab–Or contents of glasses quenched from melts coexisting with quartz, sanidine and plagioclase depend on pressure and melt water content, assuming that the normative Qtz and Ab/Or content of such melts is mainly dependent on pressure and water activity, respectively. The combination of results from the phase equilibria and from the composition of glasses indicates that the depth of magma storage for the IBR2 and BJR compositions may be in the range 300–400 MPa (~≤13 km) and 200–300 MPa (~≤10 km), respectively.

Description: Published data on Palaeogene flood basalts of the lower Mull Plateau Group (Scotland) show that the most primitive lavas (MgO 〉 8 wt %) have the greatest extent of crustal assimilation, inconsistent with a simple coupled assimilation–fractional crystallization (AFC) model. We present elemental data on rehomogenized olivine-hosted melt inclusions from four high-MgO flows to investigate the nature of crustal assimilation and melt aggregation processes during the initial stages of flood basalt magmatism on Mull. Whole-rock compositions have been variably modified by hydrothermal alteration associated with the nearby Central Complexes. Nd isotope compositions, which should be insensitive to this alteration, are lower than typical mantle values ( Nd + 2·4 to –5·7), indicating variable modification by crustal assimilation in all four samples. Melt inclusions are protected against alteration effects within their host olivine crystals, and provide more robust estimates of magmatic liquid compositions than whole-rocks, particularly for the alkali elements Na, K and Ba. The whole-rock samples show limited variations in Na 2 O (2·4–2·8 wt %) and K 2 O (0·23–0·29 wt %), despite a wide range in immobile elements (e.g. Zr 62–126 ppm). In contrast, the melt inclusions show far greater variability in Na 2 O (1·8–4·0 wt %) and K 2 O (0·02–0·35 wt %) and positive correlations between K and Na. Melt inclusions from different samples show systematic correlations between alkalis (K + Na) and incompatible element ratios (e.g. Zr/Y), indicating that the inclusions record magmatic values for the fluid-mobile elements. For the two most incompatible-element-enriched samples, the whole-rock analyses are similar to the melt inclusions except for lower Na and higher Ba that are related to alteration. Therefore, the crustal assimilation in these magmas must have taken place prior to growth of the olivines. For the two more depleted samples, the inclusions have less contaminated compositions than the whole-rocks, and show broad trends of increasing K/Ti with decreasing Fo% of the host olivine. For these samples, crustal assimilation must have taken place both during and after growth of the olivines and in an AFC style in which assimilation is linked to magmatic differentiation. Melt inclusions from single samples show limited variability in Zr/Y compared with K/Ti, indicating that aggregation of melts from different parts of the melt column must have occurred at deeper levels prior to growth of the olivines in the samples. Although the whole-rock compositional variations capture the broad details of crustal assimilation and melting histories for the Mull lavas despite the variable effects of hydrothermal alteration, the melt inclusion data more clearly resolve significant details of these magmatic processes. The extent of assimilation and differentiation is linked to the depth of magma stalling: primitive, contaminated magmas in the lower crust vs. evolved, uncontaminated magmas at sub-Moho depths.

Description: The oldest well-preserved komatiites, and the type examples, are found in the Barberton Greenstone Belt in South Africa (3·5–3·3 Ga). All three komatiite types are present, commonly within the same stratigraphic unit. Al-depleted komatiites have low Al/Ti, relatively high concentrations of incompatible elements and depleted heavy rare earth elements (HREE); Al-undepleted komatiites have chondritic Al/Ti and flat HREE patterns; and Al-enriched komatiites have high Al/Ti, low concentrations of incompatible elements, enriched HREE and extremely depleted light rare earth elements. Based on a comprehensive petrological and geochemical study, we propose a new melting model for the formation of these magmas. The basis of the model is the observation, from published experimental studies, that at great depths (~13 GPa) the density of komatiitic liquid is similar to that of solid peridotite. At such depths, melting in a rising mantle plume produces near-neutrally buoyant komatiite melt that does not escape from the residual peridotite. As the source ascends to shallower levels, however, the pressure decreases and the density difference increases, eventually making melt escape possible. Al-depleted komatiites form first at about 13 GPa by equilibrium melting under conditions in which a large proportion of melt (30–40%) was retained in the source and the residue contained a high proportion of garnet (15%). Al-undepleted and Al-enriched komatiites form by fractional melting at intermediate to shallow depths after the escape of a large proportion of melt and after exhaustion of residual garnet. This model reproduces the chemical characteristics of all komatiite types in the Barberton belt and can probably be applied to komatiites in other parts of the world.

Description: Peritectic crystals in igneous rocks may be derived from either the source or country rocks, or may have formed by reactive assimilation of source-inherited solids, primary magmatic minerals during self- or magma mixing, or country-rock xenoliths or xenocrysts. Identifying such peritectic crystals is important for constraining the components and textures of igneous rocks and the underlying processes of magmatic evolution. In this study we demonstrate that peritectic olivine formed in melting experiments crystallizes as clusters of euhedral to subhedral crystals. Olivine replacing orthopyroxene, amphibole, and phlogopite forms crystal clusters with distinct crystal to melt ratios, 2D surface area, grain boundary segmentation, and inclusion relations. In our experiments the textures of peritectic crystals are primarily controlled by the stability temperature and decomposition rate of reactive minerals. High-temperature minerals such as orthopyroxene slowly decompose to form high-density clusters of large crystals with long grain boundary segments. The SiO 2 -rich peritectic melt produced favours formation of melt inclusions. Low-temperature minerals such as amphibole and phlogopite rapidly decompose to form low-density clusters of small crystals with short grain boundary segments. The relatively SiO 2 -poor peritectic melt produced results in the formation of fewer melt inclusions. Host melt composition has a minor effect on the textures of peritectic olivine formed in the melting experiments of this study and previous contamination experiments, but affects the assemblages of the peritectic crystal clusters. Cluster density and 2D surface area of peritectic olivine tend to decrease, whereas grain boundary segment length increases with increasing experimental temperature and H 2 O content. Using textural criteria that distinguish olivine formed after different minerals in our melting experiments, we hypothesize that two olivine populations from a basaltic–andesitic lava flow of the Tatara–San Pedro volcanic complex, Chile, may be peritectic crystals formed after amphibole and orthopyroxene. Both amphibole and orthopyroxene are common in xenoliths preserved in some Tatara–San Pedro lava flows. One notable difference between the experimental and natural olivine crystals is that the natural olivine crystals have 2D surface areas and 2D grain boundary segments up to ~1000 and ~100 times larger, respectively, than those produced in our experiments. We propose that this size difference is primarily controlled by comparatively slow heating and decomposition of reactive crystals and textural coarsening of peritectic crystals during prolonged magma residence in the natural system.

Description: Lithium elemental and isotopic compositions of 33 glass and whole-rock samples from nine oceanic island regions were determined to characterize the Li inventory of the deep mantle. The Li contents of the investigated lavas range from 1·5 to 13·3 μg g – 1 , whereas 7 Li ranges from 2·4 to 4·8. There are weak co-variations between the Li/Y, 7 Li, and Sr–Nd–Pb isotope compositions of the lavas, indicating that the Li elemental and isotopic characteristics of ocean island basalt to some extent reflect mantle source heterogeneity. In detail, HIMU-type lavas are characterized by 7 Li values (up to 4·8) slightly heavier than those for average normal mid-ocean ridge basalt (3·4 ± 1·4) and by comparatively low Li contents; EM1-type lavas are characterized by isotopically light Li (average 3·2) and relative Li enrichment, whereas EM2-type lavas tend to heavier 7 Li values (up to 4·4) with high Li concentrations. The Li contents and isotope characteristics of HIMU-type lavas are consistent with recycling of altered and dehydrated oceanic crust, whereas those of the EM1-type lavas can be attributed to sediment recycling. The Li characteristics of EM2-type lavas may reflect reworking of mantle wedge material that has been infiltrated by fluids derived from the subducting plate.

Description: Magmatism in the Cenozoic Central European Volcanic Province (CEVP) has been related to two geodynamic scenarios, either extensional tectonics in the north Alpine realm or upwelling of deep mantle material. The Oligocene (~30–19 Ma) Siebengebirge Volcanic Field (SVF) is a major part of the German portion of the CEVP and consists of erosional remnants of mafic to felsic volcanic edifices. It covers an area of ~35 km (NW–SE) by ~25 km (SW–NE) with eruptive centres concentrated near the eastern shore of the Rhine river in the vicinity of the city of Bonn. Mafic rocks in the SVF comprise strongly SiO 2 -undersaturated basanites to alkaline basalts. Occurrences of alkaline basalts are confined to an inner NW–SE-striking zone, whereas the more SiO 2 -undersaturated basanites dominate the western and eastern periphery of the SVF. Radiogenic isotope compositions ( 87 Sr/ 86 Sr 0·70335–0·70371; Nd +3·1 to +4·5; Hf +6·5 to +8·0; 206 Pb/ 204 Pb 19·46–19·69; 207 Pb/ 204 Pb 15·63–15·66; 208 Pb/ 204 Pb 39·34–39·62) indicate a common asthenospheric mantle end-member with HIMU-like characteristics for all mafic rocks, similar to the European Asthenospheric Reservoir (EAR). A lithospheric mantle source component with a residual K-bearing phase (phlogopite or amphibole) is inferred from negative K anomalies. Incompatible trace element modelling indicates that melting took place in the spinel–garnet transition zone with low degrees of melting at higher pressures generating the basanitic magmas (La N /Yb N = 20–25), whereas the alkaline basalts (La N /Yb N = 14–18) are the result of higher melting degrees at shallower average melting depths. Differentiation of basanitic primary melts generated tephritic to tephriphonolitic magmas that, for instance, erupted at the Löwenburg Volcanic Complex in the central SVF. Latites and trachytes, such as the prominent Drachenfels and Wolkenburg protrusions, are more common in the central portion of the SVF. These compositions originate from parental alkaline basaltic melts. All differentiated samples show evidence for crustal contamination, possibly with lower- to mid-crustal material comprising mafic granulites as found in Eifel basalt xenoliths and metapelites. Based on the spatial and temporal distribution of the various volcanic rock types, a model for the temporal evolution of the SVF can be proposed. During the initial phase of volcanism, low-degree basanitic melts were generated as a result of decompression following tectonic rifting and formation of the Cologne Embayment, a northward extension of the Rhine Graben. In a second stage, alkali basalts were generated at shallower depths and higher degrees of melting as a result of continued lithospheric thinning and passive upwelling of asthenospheric mantle. These conclusions strengthen previous models suggesting that intraplate volcanism in Central Europe is directly linked to regional lithospheric thinning and asthenospheric upwelling. Overall, the SVF constitutes an exceptionally well-preserved magmatic assemblage to illustrate these tectono-magmatic relationships.

Description: Field, petrographic–structural and geochemical data are reported for spinel and plagioclase peridotites from the northern domain of the Lanzo peridotite massif (Western Alps). The North Lanzo peridotites are extremely heterogeneous in terms of mineral mode, texture and chemistry. They can be referred to four major groups: (1) spinel harzburgites with coarse granular to porphyroclastic structures; (2) pyroxene-depleted spinel harzburgites recording olivine-forming or pyroxene-consuming microtextures; (3) spinel lherzolites with porphyroclastic to foliated and banded structures; (4) plagioclase-enriched spinel lherzolites. Major and trace element characterization of whole-rocks and their constituent minerals allows reconstruction of the complex series of pre- to syn-rift events this mantle sector underwent. Sr, Nd and Hf isotope data provide information on the nature of infiltrating melts and time constraints. More depleted harzburgites represent refractory protoliths that after melt extraction, possibly in the presence of residual garnet, underwent a first episode of refertilization by enriched mid-ocean ridge basalt (E-MORB)-like melts, whereas harzburgites and spinel lherzolites with ocean island basalt (OIB) signatures document the successive migration of alkaline melts. The most noticeable feature of this group of rocks is their Nd–Hf decoupling, specifically the very high 176 Hf/ 177 Hf coupled with very low 143 Nd/ 144 Nd. Lu–Hf data for these peridotites define an ~260 Ma errorchron that is interpreted as evidence of mixing during relatively recent times between old (most probably Proterozoic) refractory subcontinental mantle and OIB-type melts. This event most probably occurred during extension (Triassic times) or during the onset of exhumation (Liassic times). Plagioclase peridotites document the last refertilization episode, involving the shallow-level impregnation of harzburgite mantle by evolved MORB melts before its sea-floor emplacement. This Middle Jurassic event caused the almost complete resetting of the original trace element and 87 Sr/ 86 Sr composition. The combination of structural, petrological and geochemical information for a north–south Lanzo traverse, from the North Massif to the South Massif, notwithstanding the effects of the alpine orogeny, allows the study of the complete evolution of a sector of old (Proterozoic?) mantle since the early stages of melt removal and allows reconstruction of the tectonic and magmatic events during continental extension leading to the opening of the Jurassic Ligurian–Piedmontese basin. North Lanzo fundamentally preserves the record of pre-syn-rift ancient episodes, whereas South Lanzo better highlights the processes that deeply modified and refertilized the older lithosphere during subsequent lithosphere extension. Slow to very slow extension led to sea-floor exposure of the subcontinental lithospheric mantle (North Lanzo) at a marginal position, close to the Adria continental margin, and of the deeply melt-modified lithospheric mantle (Lanzo South) in a more distal setting of the basin. In this respect, the Lanzo traverse is closely similar to the ocean–continent transition in slow- or ultraslow-spreading oceanic basins, such as the North Atlantic.

Description: Basanites from the Tertiary Siebengebirge area of Germany (part of the Central European Volcanic Province; CEVP) have high Mg# (〉0·60), moderate to high Cr (〉300 ppm) and Ni (〉200 ppm) contents and strong light rare earth element enrichment, but systematic depletion in Rb and K relative to trace elements of similar compatibility in anhydrous mantle. Rare earth element melting models can explain the petrogenesis of these basanites in terms of partial melting of a spinel peridotite source containing residual amphibole. It is inferred that amphibole, indicated by the relative K and Rb depletion and the melting model, was precipitated in the spinel peridotite lithospheric mantle beneath the Siebengebirge, by metasomatic fluids or melts from a rising mantle diapir or plume. Alkali basalts and more differentiated rocks have lower Mg# and lower abundances of Ni and Cr, and have undergone fractionation of mainly olivine, clinopyroxene, Fe–Ti oxides, amphibole and plagioclase. Most of the basanites and alkali basalts approach the Sr–Nd–Pb isotope compositions inferred for the European Asthenospheric Reservoir component. Trace element constraints (i.e. low Nb/U and Ce/Pb ratios) and the Sr–Nd–Pb isotope composition of the differentiated rocks indicate that assimilation of lower crustal material has modified the composition of the primary mantle-derived magmas. High 207 Pb/ 204 Pb ratios in the differentiated lavas point to assimilation of ancient lower crustal components having high U/Pb and Th/Pb ratios. Relatively shallow melting of inferred amphibole-bearing spinel peridotite sources may suggest an origin from the metasomatized part of the thermal boundary layer. Application of new thermobarometric equations for the basaltic magmas indicates relatively normal mantle potential temperatures (1300–1400°C); thus the inferred mantle ‘baby plume’ or ‘hot finger’ is not thermally anomalous.

Description: We consider an electromagnetic scattering problem produced by a perfect conductor. We pose the problem in a bounded region surrounding the obstacle and impose on the exterior boundary of the computational domain an impedance boundary condition inspired by the asymptotic behaviour of the scattered field at infinity. The operator associated with our problem belongs to a class of operators for which a suitable decomposition of the energy space plays an essential role in the analysis. This decomposition is performed here through a regularizing projector that takes into account the boundary conditions. The discrete version of this projector is the key tool to prove that a Galerkin scheme based on Nédélec’s edge elements is well posed and convergent under general topological assumptions on the scatterer and without assuming special requirements on the triangulations.

Description: Gould and Robinson (2010, SIAM J. Optim. , 20 , 2023–2048; 2010, SIAM J. Optim. , 20 , 2049–2079) introduced a second-derivative sequential quadratic programming method (S2QP) for solving nonlinear nonconvex optimization problems. We proved that the method is globally and locally superlinearly convergent under common assumptions. A critical component of the algorithm is the so-called predictor step, which is computed from a strictly convex quadratic program with a trust-region constraint. This step is essential for proving global convergence but its propensity to identify the optimal active set is paramount for achieving fast local convergence. Thus the global and local efficiency of the method is intimately coupled with the quality of the predictor step. In this paper we study the effects of removing the trust-region constraint from the computation of the predictor step. This is reasonable since the resulting problem is still strictly convex and thus well defined. Although it is interesting theoretically to verify that the same convergence guarantees hold when no trust-region constraint is used, our motivation is based on the practical behaviour of the algorithm. Preliminary numerical experience with S2QP indicates that the trust-region constraint occasionally degrades the quality of the predictor step and diminishes its ability to correctly identify the optimal active set. Moreover, removal of the trust-region constraint allows for re-use of the predictor step over a sequence of failed iterations, thus reducing computation. We show that the modified algorithm remains globally convergent and preserves local superlinear convergence provided that a nonmonotone strategy is incorporated.

Description: We consider a recently proposed finite-element space that consists of piecewise affine functions with discontinuities across a smooth given interface (a curve in two dimensions, a surface in three dimensions). Contrary to existing extended finite element methodologies, the space is a variant of the standard conforming $${\mathbb{P}}_{1}$$ space that can be implemented element by element. Further, it neither introduces new unknowns nor deteriorates the sparsity structure. It is proved that, for u arbitrary in $${W}^{1,p}(\Omega \setminus \Gamma )\cap {W}^{2,s}(\Omega \setminus \Gamma )$$ , the interpolant $${\mathcal{I}}_{h}u$$ defined by this new space satisfies where h is the mesh size, $$\Omega \subset {\mathbb{R}}^{d}$$ is the domain, $$p 〉 d$$ , $$p\ge q$$ , $$s\ge q$$ and standard notation has been adopted for the function spaces. This result proves the good approximation properties of the finite-element space as compared to any space consisting of functions that are continuous across , which would yield an error in the $${L}^{q}(\Omega )$$ -norm of order . These properties make this space especially attractive for approximating the pressure in problems with surface tension or other immersed interfaces that lead to discontinuities in the pressure field. Furthermore, the result still holds for interfaces that end within the domain, as happens for example in cracked domains.

Description: A linear parabolic differential equation on a moving surface is first discretized in space by evolving surface finite elements and then in time by an implicit Runge–Kutta (RK) method. For algebraically stable and stiffly accurate RK methods unconditional stability of the full discretization is proven and the convergence properties are analysed. Moreover, the implementation is described for the case of the Radau IIA time discretization. Numerical experiments illustrate the behaviour of the fully discrete method.

Description: The build-up of large magmatic complexes can proceed piecemeal over periods of several million years through sequences of complex processes of magma production, differentiation, assimilation, final crystallization and subsequent metasomatic modification. All these stages can produce or modify minerals used as geochronometers, such as zircon, monazite and titanite. The present study exemplifies such complex relationships, also demonstrating how a systematic approach with comprehensive sampling and careful high-resolution U–Pb analyses can yield a coherent picture of the entire magmatic process. The study was conducted on the Pavia pluton, an elongated Variscan intrusion in the Ossa–Morena Zone of Portugal. The geochronological data show that the Pavia pluton was emplaced by the amalgamation of multiple magma pulses into the crust, over a period of c . 11 Myr. An early event at ~340 Ma, revealed by xenocrystic zircon, preceded the magmatic activity at the exposed level of the pluton, but is recognized as the main magmatic event elsewhere in the Ossa–Morena Zone. A second event at 328 Ma formed tonalite, trondhjemite and granodiorite, and subordinate differentiates in the central domains of the pluton (units I and II). A third event at c . 324 Ma emplaced granodiorite in the flanking domains III–V and the contemporaneous and widespread two-mica granite in domain VI, together with late rhyodacite porphyries, microgranodiorites, aplite–pegmatite and pegmatite dikes. A fourth event at 319–317 Ma was characterized by the emplacement of some microgranites and pegmatite dikes. These two last magmatic events also had an effect on the previously emplaced rocks, causing local overgrowths and isotopic resetting of minerals. The occurrence of a fifth magmatic event at depth at 313 Ma is the inferred cause of the hydrothermal activity responsible for local zircon, monazite and titanite resorption and/or recrystallization and for some of the textures exhibited by the main rock-forming minerals. The magmatic episodes were interspersed with periods of quiescence; this cyclicity presumably reflects an external control by the transtensional tectonic regime of the Ossa–Morena Zone.

Description: Igneous rock textures reflect the cooling history of the parental magma. Combined with chemical data, they can provide physical and chemical information about the evolution of a magma body. The petrographic textures and chemical compositions of 21 coarse- and fine-grained granite samples along an ~250 m horizontal outcrop of the Shanggusi granite porphyry are presented in this case study. The coarse-grained granite porphyry is an early intrusion, and the fine-grained granite dykes, mostly intruded into the granite porphyry, are later intrusions. The studied samples have nearly homogeneous major element bulk-rock and mineral compositions, but show large variations in their trace element compositions and textural characteristics. The trace element data suggest the influence of hydrous fluids (possibly enriched in CO 2 , F, and Cl) in the evolution of the plutonic body. Textural analysis of the coarse-grained granite porphyry indicates that the crystal size distribution (CSD) slopes, intercepts and total numbers of groundmass decrease from the center to the margin of the intrusion in contrast to the maximum diameter of the crystals (L max ) (average length of the four largest quartz crystals for each sample); however, most fine-grained samples and the groundmass of the coarse-grained samples show concave-down CSDs, indicating textural coarsening. Quartz CSDs in the coarse-grained samples are kinked, with a steep-sloped log–linear section representing small crystals (〈1 mm) and a shallow-sloped log–linear section representing large crystals (〉1 mm). These two crystal populations are interpreted as resulting from a shift in cooling regime. The straight CSDs of two fine-grained samples may be due to a different cooling history. In general, the spatial variation of the CSD patterns can be attributed to various degrees of overgrowth and mechanical compaction. The quartz phenocrysts in several coarse-grained samples exhibit a high degree of alignment, which may be the result of magmatic flow. By integrating the field geology, geochemistry and quantitative textural data from the horizontal profile of the Shanggusi granite porphyry, it is suggested that hydrous fluids at the top of the intrusion not only controlled the fractionation of elements but also affected its cooling history. Fluid migration-controlled undercooling can explain the solidification processes in the Shanggusi intrusion, and may also be prevalent in other fluid-rich shallow intrusions. Quantitative integration of textural and geochemical data for igneous rocks can contribute to our understanding of the relationships between physical and chemical processes in a magma system, and provide relatively comprehensive insights into the petrogenesis of granites.

Description: Cenozoic volcanism within Mongolia forms part of a large central Asian province of intra-plate magmatism. Numerous small-volume volcanic cones and alkali basalt lava flows have been formed since c . 30 Ma; from c . 12 Ma activity has been focused on the uplifted Hangai dome. A mechanism for melting beneath the dome has, however, thus far remained enigmatic. Some of the oldest basalts on the Hangai dome erupted at its centre at ~6 Ma and their geochemistry suggests a garnet lherzolite source region at 90–100 km depth. These lavas have Pb isotope compositions similar to those of depleted Indian mid-ocean ridge basalts (MORB) ( 206 Pb/ 204 Pb = 17·822, 207 Pb/ 204 Pb = 15·482, 208 Pb/ 204 Pb = 37·767), which may be indicative of the involvement of ambient asthenospheric mantle in their petrogenesis. Younger basalts exhibit a gradual shift in isotopic composition towards a source that has less radiogenic Pb and more radiogenic Sr, evidenced by the eruption of lavas with 206 Pb/ 204 Pb = 16·991 and 87 Sr/ 86 Sr = 0·704704. The youngest lavas, dated as younger than ~8 ka, have the highest K 2 O contents (up to 5·2 wt %) and are characterized by the most enriched trace-element signatures; they are interpreted to represent melting of a metasomatically altered sub-continental lithospheric mantle containing phlogopite. Concurrent with progressive melting of the lithosphere, melting appears to propagate outwards from the centre of the dome to its margins; by 0·7 Ma the marginal magmatism is interpreted to result from melting of a depleted MORB-source mantle component with a smaller contribution from the lithospheric mantle. The spatial and temporal variations in melting beneath the Hangai dome may be explained by either lithospheric delamination or the presence of a small-scale thermal anomaly in the upper mantle. Although it is not possible to distinguish between these models on the basis of geochemistry alone, the lack of a viable mechanism to generate small-scale upwelling lends support to a model involving delamination of the lithospheric mantle beneath the Hangai dome.

Description: The Hidaka Metamorphic Belt, in southeastern Hokkaido, Japan, provides insights into how magmatic sulfide deposits may form through magma mixing deep within arcs. Here, a near-complete cross-section of arc crust is exposed, with large mafic igneous complexes preserved at deeper levels. Magmatic sulfide mineralization occurs within the Opirarukaomappu Gabbroic Complex (OGC), which preserves a record of crustal contamination of mafic magmas via assimilation and magma mixing involving introduction of crust-derived tonalite. Assimilation–fractional crystallization modelling suggests that the gabbro evolved through a combination of approximately 10 wt % mixing and 14 wt % fractional crystallization. Magmatic sulfides and associated gabbros, diorites and tonalites at this locality contain graphite, with carbon isotope signatures consistent with derivation from the surrounding partially melted carbonaceous shales. This indicates that crust-derived carbon was added to the mafic magma through the magma mixing and assimilation process. Sulfur isotope data suggest that sulfur was also added from crustal sources during assimilation and magma mixing. The relationships observed in the OGC suggest that intrusion of basalt into a segment of deep arc crust drove partial melting of carbonaceous metamorphic rocks, producing graphite-bearing felsic magmas with high reducing potential. Redox budget modelling shows that mixing of only small proportions of these magmas is sufficient to lower the oxidation state of oxidized basaltic magmas enough to induce sulfide saturation and consequent exsolution of immiscible sulfide melt. Magmatic sulfide deposits are likely to form by this reduction-induced sulfide saturation mechanism deep within other arcs where magma mixing is thought to be common.

Description: A reactive flow geochemical model based on pMELTS thermodynamic calculations explains the observed modal, major, and trace element variations in the Red Hills peridotite, New Zealand. The model also reproduces the major and trace element chemical variations in mid-ocean ridge basalts (MORB) observed in present-day spreading ridges. The Red Hills peridotite is thought to originate from palaeo-MOR magmatic processes in the mantle–Moho transition zone. The peridotite body consists of a harzburgite matrix and dunite channels. The harzburgite forms the Lower Unit and is intruded by replacive dunite channels in the Upper Unit. This lithology gradually turns into a massive dunite zone in which disseminated to lenticular clinopyroxene aggregates are present. The rare earth element (REE) abundances in the peridotite samples vary greatly depending on their lithologies. In the Lower Unit, REE are extremely depleted, whereas in the Upper Unit they are relatively enriched, in contradiction to the depleted lithologies. Our model consists of two stages. The first stage assumes melting of depleted MORB source mantle in the garnet stability field, and the second assumes reactions between residual solids and the melts from the first stage in the spinel stability field in an open system. The model explains the formation of depleted harzburgite and the formation of dunite channels in the harzburgite matrix well. The major and trace element compositions of the melts calculated by the model vary from ultra-depleted MOR melts in harzburgite to normal MORB in dunite, suggesting that these lithologies are residues of a palaeo-MOR. The model also explains the origins of the local and global geochemical trends observed in MORB and the geochemical variation in abyssal peridotite samples. Our model confirms the important role of reactive flow in the mantle–Moho transition zone beneath MORs.

Description: The islands of Flores and Corvo in the Azores archipelago are the only two of nine subaerial volcanic edifices lying west of the Mid-Atlantic Ridge (MAR). This makes them important for constraining the evolution of this young (〈40 Ma) oceanic plateau. The alkalic basalt suites from Flores and Corvo lie on a single liquid line of descent. Ankaramitic cumulates, with MgO contents up to ~18 wt %, result from clinopyroxene-dominated polybaric crystallization. The parental magmas (MgO ~ 11 wt %) are inferred to be low-degree partial melts ( F = 3–5%) of enriched peridotite generated at depths of ~80–90 km. These primary magmas commenced crystallizing at the lithosphere–asthenosphere boundary and this continued in conduits over a pressure range of ~0·6–1·2 GPa. Only lavas with MgO 〈 3 wt % fractionated at shallow crustal levels. Nd and Sr isotope data reveal variations in the source of both magmatic systems, suggesting variable contributions from both enriched (E-) and depleted (D-) mid-ocean ridge basalt (MORB)-source mantle components. This is supported by the greater variability of incompatible trace-element ratios within the Flores lavas (e.g. Ba/Nd, La/Sm, Th/Nd), whereas those from Corvo exhibit a good correlation between key trace-element ratios [e.g. (La/Sm) N , Th/Nd] and Sr isotope ratios. Lavas from Flores display a greater variability in Sr and Nd isotope compositions and define a mixing array between an E-MORB source and a common Azores mantle source. The latter signature is restricted to lava suites from the north and east of Flores. We concur with the generally accepted notion that Flores and Corvo are derived from the same mantle plume as is responsible for the eastern Azores islands. However, there is evidence (different Nb/Zr, Ta/Hf and La/Sm, but homogeneous Sr and Nd isotopic composition) that these two islands are dominated by a source component that is not as evident in the eastern archipelago.

Description: Significant differences between bulk-rock lithophile trace element budgets and the sum of the contributions from the constituent minerals are common, if not ubiquitous, in peridotite xenoliths. Notwithstanding the possible contributions from fluid inclusions and grain-boundary glass and micro-phases, it is often difficult to reconcile the bulk-rock incompatible element budgets with those of the silicate phases. In the absence of modal metasomatism this discrepancy is often attributed to the ‘catch-all’ yet often vague process of cryptic metasomatism. This study presents comprehensive petrological descriptions, major and trace element and Sr–Nd isotope data for variably metasomatized bulk-rock peridotites, host basalt, and constituent peridotite mineral phases from spinel lherzolite and harzburgite xenoliths from the Kilbourne Hole volcanic maar, New Mexico, USA. Similar measurements were also made on hand-picked interstitial glass from one of the most highly metasomatized samples in an attempt to unravel the sources, effects, and relative timings of multiple metasomatic events. Reaction textures around clinopyroxene grains are evident in the most metasomatized samples. These are accompanied by films of high-SiO 2 interstitial glass, which is not in equilibrium with the primary silicate minerals. Trails of glassy melt inclusions terminate in these films against which the margins of the primary minerals appear partially resorbed. Furthermore, different styles of high field strength element fractionation [e.g. (Nb/Ta) N vs (Zr/Hf) N ] are evident in the bulk-rocks and the clinopyroxenes that they host. In all of the Kilbourne Hole peridotites analysed, hand-picked, optically clean clinopyroxenes preserve a more unradiogenic Sr isotope signature than the corresponding bulk-rock. Hand-picked interstitial glass from KH03-16 reveals the most radiogenic 87 Sr/ 86 Sr of any component recovered from these xenoliths ( 87 Sr/ 86 Sr = 0·708043 ± 0·00009; [Sr] = 81 ppm). Similarly, the 143 Nd/ 144 Nd of the glass is amongst the most evolved of the peridotite components ( 143 Nd/ 144 Nd = 0·512893 ± 0·000012; [Nd] = 10 ppm). However, the host basalt ( 87 Sr/ 86 Sr = 0·703953 ± 0·00012; 143 Nd/ 144 Nd = 0·512873 ± 0·000013), similar in composition to that of the nearby, contemporaneous, Potrillo Volcanic Field basalts, contains nearly an order of magnitude more Sr and more than three times more Nd ([Sr] = 655 ppm; [Nd] = 34 ppm) than the interstitial glass. Despite the high Sr and Nd contents of the host basalt the evidence for basalt infiltration is scant, although the effects of melt–rock interaction, both in antiquity and more recently, are preserved in several xenoliths. Mixing between clinopyroxene and the host basalt cannot account for the full range of bulk-rock Sr–Nd isotope ratios; nearly half of the xenoliths require an additional component that could involve varying amounts of interstitial glass. In detail the behaviour of Sr and Nd isotopes has been decoupled, requiring multiple, temporally distinct, metasomatic events. Several bulk-rock samples require a further, as yet unidentified, component to explain the bulk-rock trace element mass balance and Sr–Nd isotope composition fully, implying that at least three episodes of melt–rock interaction, refertilization and metasomatism must have occurred prior to the arrival of the xenoliths at the surface in their host maar deposits.

Description: Isothermal and isobaric crystallization of plagioclase in a water-saturated synthetic rhyolitic melt is investigated through a time-series of decompression experiments. The experimental variables are the rate at which samples are initially decompressed (30, 150, and 1200 MPa h – 1 ) from 200 MPa and 875°C, final pressure (25–160 MPa), and holding time at final pressure (up to 17 days). Through textural measurements of the crystals, plagioclase crystallization kinetics is characterized in terms of nucleation lag and rates of nucleation and growth. Plagioclase crystallization is markedly dependent on effective undercooling, T eff , and holding time at crystallization pressure. With T eff increasing from 55 to 110°C, (1) nucleation lag decreases from 1–2 days to ~15 min, (2) maximum nucleation rates increase from ~10 – 3 to 10 – 2 mm – 2 s – 1 , and (3) maximum growth rates decrease from ~10 – 6 to 5 x 10 – 7 mm s – 1 . The initial decompression rate (30, 150, and 1200 MPa h – 1 ) has no systematic control on crystallization at final pressure, except for the 1200 MPa h – 1 series in which samples show nucleation difficulties. From the experimental data for T eff -constrained plagioclase number density, proportion, and morphology, we provide means to assess the conditions of nucleation and growth of natural plagioclase microlites from rapidly ascended rhyolitic melts, through the determination of the plagioclase liquidus curve and T eff prevailing during crystallization.

Description: New geochemical and isotopic data are presented for volumetrically minor, depleted low-Ti basalts that occur in the Plateau Basalt succession of central East Greenland (CEG), formed during the initial stages of opening of the North Atlantic at 55 Ma. The basalts have mid-ocean ridge basalt (MORB)-like geochemistry (e.g. depleted light rare earth elements) and are distinct from the high-Ti lavas that dominate the sequence. Rare earth element geochemistry implies derivation from a source more depleted than the typical MORB source, and suggests polybaric melting and contributions from both spinel- and garnet-facies mantle. The low-Ti basalts have Sr–Nd–Pb–Hf isotopic characteristics that are similar to those of depleted magmas from Iceland (e.g. Theistareykir) and adjacent ridges (Kolbeinsey and Reykjanes) and distinct from global MORB (e.g. negative 207 Pb, and Hf and Nd isotope compositions that plot above the mantle reference line). Isotope and trace element data indicate the involvement of two depleted source components. One component has isotopic compositions similar to other depleted components identified in the North Atlantic and has high Rb/Zr and Ba/Nb. The second is isotopically less depleted with lower Rb/Zr and Ba/Nb. Small degrees of crustal contamination (〈 1%) by both amphibolitic and granulitic crust result in relatively large changes in isotopic composition ( c . 1% lower for 206 Pb/ 204 Pb and 0·1% higher for 87 Sr/ 86 Sr depending on the contaminant). Negative Nb suggests a MORB affinity for the low-Ti magmas; however, they are distinguished from global normal (N)-MORB on the basis of vertical deviations from the Northern Hemisphere Reference Line (negative 207 Pb and positive 208 Pb), and relative enrichments in Ba, Sr and Pb. The isotopic compositions of the low-Ti CEG basalts suggest correlation with modern depleted components beneath Iceland and adjacent ridges, considered to be derived from upper mantle sources polluted by the Iceland plume. However, small positive Pb peaks when normalized to MORB, and lower Nb distinguish the CEG low-Ti basalts from depleted Icelandic compositions. The lower Nb (〈 0) and 87 Sr/ 86 Sr, and suggestion of higher 206 Pb/ 204 Pb in crustally uncontaminated parental melts imply a closer affinity to compositions from the oceanic ridges surrounding Iceland (especially the Reykjanes Ridge), yet they are subtly distinct on the basis of available trace element data. We suggest that this depleted component was an integral part of the plume that melted primarily during the rapid lithospheric uplift and extension associated with continental break-up.

Description: The origin and evolution of diamondiferous lithospheric mantle sampled by the Neoproterozoic Renard kimberlites (eastern Superior Province, Quebec) is constrained based on mantle-derived microxenoliths and xenocrysts. The dataset illustrates the wealth of knowledge that can be gleaned from small samples (1·2 mg–2·2 g) through an integration of multiple, mainly single mineral based approaches. Our samples document the presence of an ~200 km thick lithosphere with a ‘cold’ (38 mW m – 2 surface heat flow) model geotherm at the time of kimberlite emplacement ( c . 632 Ma), resulting in a large diamond window from 130 to 200 km (42–60 kbar). On the basis of the mantle xenolith and xenocryst record and excluding megacrysts, the lithospheric mantle beneath Renard was dominated by peridotite (91%), composed of lherzolite (72% of samples), harzburgite (24%) and wehrlite (5%), with minor eclogitic (3%) and websteritic (6%) portions. Comparatively abundant harzburgite probably establishes the principal diamond source, but elevated Na contents in eclogitic garnet suggest the additional presence of diamond-stable eclogites. A number of events have modified the lithospheric mantle underlying the eastern Superior Province, including the following: (1) evolving ‘kimberlitic’ melts pervasively re-fertilized the originally strongly depleted lithospheric mantle with respect to highly and moderately incompatible trace elements; (2) less pervasive fluid style metasomatism is indicated by selective re-enrichment of highly incompatible elements that occurred within a depth range of 125–170 km. In situ Pb isotope data obtained for clinopyroxenes suggest a model age of ~2·7 Ga for the protolith(s) of the cratonic lithospheric mantle beneath Renard. This age coincides with a major phase of continental crust generation within the Superior Province and throughout the Laurentia supercontinent (e.g. Greenland).

Description: In this paper we develop the a priori and a posteriori error analyses of a mixed finite element method for the coupling of fluid flow with nonlinear porous media flow. Flows are governed by the Stokes and nonlinear Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces and the Beavers–Joseph–Saffman law. We consider dual-mixed formulations in both domains, and in order to handle the nonlinearity involved, we introduce the pressure gradient in the Darcy region as an auxiliary unknown. In addition, the transmission conditions become essential, which leads to the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. As a consequence, the resulting variational formulation can be written, conveniently, as a twofold saddle point operator equation. Thus, a well-known generalization of the classical Babuska–Brezzi theory is applied to show the well-posedness of the continuous and discrete formulations and to derive the corresponding a priori error estimate. In particular, the set of feasible finite element subspaces includes Raviart–Thomas elements of lowest order and piecewise constants for the velocities and pressure, respectively, in both domains, together with piecewise constant vectors for the Darcy pressure gradient and continuous piecewise linear elements for the traces. Then, we employ classical approaches and use known estimates to derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. Finally, several numerical results confirming the good performance of the method and the theoretical properties of the a posteriori error estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution, are reported.

Description: We consider a piecewise linear, discontinuous Galerkin method for the time discretization of a fractional diffusion equation involving a parameter in the range –1 〈 α 〈 0. Our analysis shows that, for a time interval (0, T ) and a spatial domain , the uniform error in L ((0, T ); L 2 ( )) is of order k , where = min g (2, +α) and k denotes the maximum time step. Thus, if –1/2 ≤ α 〈 0, then we have optimal O( k 2 ) convergence, just as for the classical diffusion (heat) equation.

Description: We propose a mixed finite-element method for the motion of a strongly viscous, ideal and isentropic gas. At the boundary we impose a Navier slip condition such that the velocity equation can be posed in mixed form with the vorticity as an auxiliary variable. In this formulation we design a finite-element method, where the velocity and vorticity are approximated with the div- and curl-conforming Nédélec elements, respectively, of the first order and first kind. The mixed scheme is coupled to a standard piecewise constant upwind discontinuous Galerkin discretization of the continuity equation. For the time discretization implicit Euler time stepping is used. Our main result is that the numerical solution converges to a weak solution as the discretization parameters go to zero. The convergence analysis is inspired by the continuous analysis of Feireisl and Lions for the compressible Navier–Stokes equations. Tools used in the analysis include an equation for the effective viscous flux and various renormalizations of the density scheme.

Description: This paper aims to present a unified framework for deriving analytical formulas for smoothing factors in arbitrary dimensions, under certain simplifying assumptions. To derive these expressions we rely on complex analysis and geometric considerations, using the maximum modulus principle and Möbius transformations. We restrict our attention to pointwise and block lexicographic Gauss–Seidel smoothers on a d -dimensional uniform mesh, where the computational molecule of the associated discrete operator forms a (2 d +1)-point star. In the pointwise case, the effect of a relaxation parameter is analysed. Our results apply to any number of spatial dimensions and are applicable to high-dimensional versions of a few common model problems with constant coefficients, including the Poisson and anisotropic diffusion equations, as well as a special case of the convection–diffusion equation. We show that in most cases our formulas, exact under the simplifying assumptions of local Fourier analysis, form tight upper bounds for the asymptotic convergence of geometric multigrid in practice. We also show that there are asymmetric cases where lexicographic Gauss–Seidel smoothing outperforms red–black Gauss–Seidel smoothing; this occurs for certain model convection–diffusion equations with high mesh Reynolds numbers.

Description: We study the stability properties of, and the phase error present in, several higher-order (in space) staggered finite difference schemes for Maxwell's equations coupled with a Debye or Lorentz polarization model. We present a novel expansion of the finite difference approximations, of arbitrary (even) order, of the first-order spatial derivative operator. This alternative representation allows the derivation of a concise formula for the numerical dispersion relation for all even order schemes applied to each model, including the limiting (infinite-order) case. We further derive a closed-form analytical stability condition for these schemes as a function of the order of the method. Using representative numerical values for the physical parameters, we validate the stability criterion while quantifying numerical dissipation. Lastly, we demonstrate the effect that the spatial discretization order, and the corresponding stability constraint, has on the dispersion error.

Description: The aim of this paper is to analyse a numerical method to solve transient eddy current problems with input current intensities as data, formulated in terms of the magnetic field in a bounded domain including conductors and dielectrics. To this end, we introduce a time-dependent weak formulation and prove its well-posedness. We propose a finite element method for space discretization based on the Nédélec edge elements on tetrahedral meshes, for which we obtain error estimates. Then we introduce a backward Euler scheme for time discretization and prove error estimates for the fully discrete problem, too. Furthermore, a magnetic scalar potential is introduced to deal with the curl-free condition in the dielectric domain, which leads to an important saving in computational effort. Finally, the method is applied to solve two problems: a test with a known analytical solution and an application to electromagnetic forming.

Description: In this paper a quadrature method for Cauchy singular integral equations having constant coefficients and index equal to –1 is proposed. A polynomial approximation of the solution is constructed by solving a determined and well-conditioned linear system. Error estimates and numerical tests are also included.

Description: We introduce a novel technique for constructing higher-order variational integrators for Hamiltonian systems of ordinary differential equations. In the construction of the discrete Lagrangian we adopt Hermite interpolation polynomials and the Euler–Maclaurin quadrature formula and apply collocation to the Euler–Lagrange equation and its prolongation. Considerable attention is devoted to the order analysis of the resulting variational integrators in terms of approximation properties of the Hermite polynomials and quadrature errors. In particular, the order of the variational integrator can be computed a priori based on the quadrature error estimate. The analysis in the paper is straightforward compared to the order theory for Runge–Kutta methods. Finally, a performance comparison is presented on a selection of these integrators.

Description: Linear systems of equations Ax = b , where the matrix A has some particular structure, arise frequently in applications. Very often, structured matrices have huge condition numbers and, therefore, standard algorithms fail to compute accurate solutions of Ax = b . We say in this paper that a computed solution is accurate if being the unit roundoff. In this work we introduce a framework that allows many classes of structured linear systems to be solved accurately, independently of the condition number of A and efficiently, that is, with cost For most of these classes no algorithms are known that are both accurate and efficient. The approach in this work relies on first computing an accurate rank-revealing decomposition of A , an idea that has been widely used in the last decades to compute singular value and eigenvalue decompositions of structured matrices with high relative accuracy. In particular, we illustrate the new method by accurately solving Cauchy and Vandermonde linear systems with any distribution of nodes, that is, without requiring A to be totally positive for most right-hand sides b .

Description: This paper is concerned with a staggered discontinuous Galerkin method for the curl–curl operator arising from the time-harmonic Maxwell equations. One distinctive feature of the method is that the discrete operators preserve the properties of the differential operators. Moreover, the numerical solution automatically satisfies a discrete divergence-free condition. Stability and optimal convergence of the method are analysed. Numerical experiments for smooth and singular solutions are shown to verify the optimal order of convergence. Furthermore, the method is applied to the corresponding eigenvalue problem. Numerical results for rectangular and L-shaped domains show that our method is able to produce nonspurious eigenvalues.

Description: We propose a general framework that allows for a new natural coupling of boundary element and a wide class of finite element methods (FEMs) for a model second-order elliptic problem. This class of FEMs includes mixed methods, discontinuous Galerkin methods and the continuous Galerkin method. We provide sufficient conditions guaranteeing the well-posedness of the methods and give several examples that include new as well as old methods.

Description: The Lippmann–Schwinger integral equation describes the scattering of acoustic waves from an inhomogeneous medium. For scattering problems in free space, Vainikko proposed a fast spectral solution method exploiting the convolution structure of this equation's integral operator and the fast Fourier transform. Although the integral operator of the Lippmann–Schwinger integral equation for scattering in a planar three-dimensional waveguide is not a convolution, we show in this paper that the separable structure of the kernel allows to construct fast spectral collocation methods. The numerical analysis of this method requires smooth material parameters; for discontinuous materials there is no theoretical convergence statement. Therefore, we construct a Galerkin variant of Vainikko's method avoiding this drawback. For several distant scattering objects inside the three-dimensional waveguide this discretization technique would lead to a computational domain consisting of one large box containing all scatterers and hence many unnecessary unknowns. However, the integral equation can be reformulated as a coupled system with unknowns defined on the different parts of the scatterer. Discretizing this coupled system by a combined spectral/multipole approach yields an efficient method for waveguide scattering from multiple objects.

Description: The aim of this paper is to investigate the stability of time integration schemes for the solution of a finite element semi-discretization of a scalar convection–diffusion equation defined on a moving domain. An arbitrary Lagrangian–Eulerian formulation is used to reformulate the governing equation with respect to a moving reference frame. We devise an adaptive -method time integrator that is shown to be unconditionally stable and asymptotically second-order accurate for smoothly evolving meshes. An essential feature of the method is that it satisfies a discrete equivalent of the well-known geometric conservation law. Numerical experiments are presented to confirm the findings of the analysis.

Description: Stochastic collocation methods facilitate the numerical solution of partial differential equations (PDEs) with random data and give rise to long sequences of similar linear systems. When elliptic PDEs with random diffusion coefficients are discretized with mixed finite element methods in the physical domain we obtain saddle point systems. These are trivial to solve when considered individually; the challenge lies in exploiting their similarities to recycle information and minimize the cost of solving the entire sequence. We apply stochastic collocation to a model stochastic elliptic problem and discretize in physical space using Raviart–Thomas elements. We propose an efficient solution strategy for the resulting linear systems that is more robust than any other in the literature. In particular, we show that it is feasible to use finely-tuned algebraic multigrid preconditioning if key set-up information is reused. The proposed solver is robust with respect to variations in the discretization and statistical parameters for stochastically linear and nonlinear data.

Description: We address numerically the question of the asymptotic stability of equilibria for a Gurtin–MacCamy model with age-dependent spatial diffusion. The problem reduces to the study of a finite number of simpler models without diffusion, which are parametrized by the eigenvalues of the Laplacian operator. Here the approach in Breda et al. (2007, Stability analysis of age-structured population equations by pseudospectral differencing methods. J. Math. Biol. , 54 , 701–720; 2008, Stability analysis of the Gurtin–MacCamy model. SIAM J. Numer. Anal. , 46 , 980–995), which is based on pseudospectral methods, is adapted to the reduced models and the error analysis is revisited in order to prove the preservation of convergence of infinite order.

Description: The moving least squares (MLS) method provides an approximation û of a function u based solely on values u ( x j ) of u on scattered ‘meshless’ nodes x j . Derivatives of u are usually approximated by derivatives of û . In contrast to this, we directly estimate derivatives of u from the data, without any detour via derivatives of û . This is a generalized MLS technique, and we prove that it produces diffuse derivatives as introduced by Nyroles et al. (1992, Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. , 10 , 307–318). Consequently, these turn out to be efficient direct estimates of the true derivatives, without anything ‘diffuse’ about them, and we prove optimal rates of convergence towards the true derivatives. Numerical examples confirm this, and we finally show how the use of shifted and scaled polynomials as basis functions in the generalized and standard MLS approximation stabilizes the algorithm.

Description: The differential linear variational inequality consists of a system of n ordinary differential equations (ODEs) and a parametric linear variational inequality as the constraint. The right-hand side function in the ODEs is not differentiable and cannot be evaluated exactly. Existing numerical methods provide only approximate solutions. In this paper we present a reliable error bound for an approximate solution x h ( t ) delivered by the time-stepping method, which takes all discretization and roundoff errors into account. In particular, we compute two trajectories x j h ( t )± j h ( t ) to determine the existence region of the exact solution for each . Moreover, we have . Numerical examples of bridge collapse, earthquake-induced structural pounding and circuit simulation are given to illustrate the efficiency of the error bound.

Description: We present a comprehensive convergence analysis for discontinuous piecewise polynomial approximations of a first-kind Volterra integral equation with smooth convolution kernel, examining the attainable order of (super-) convergence in collocation, quadrature discontinuous Galerkin (QDG) and full discontinuous Galerkin (DG) methods. We introduce new polynomial basis functions with properties that greatly simplify the convergence analysis for collocation methods. This also enables us to determine explicit formulae for the location of superconvergence points (i.e., discrete points at which the convergence order is one higher than the global bound) for all convergent collocation schemes. We show that a QDG method, which is based on piecewise polynomials of degree m and uses exactly m + 1 quadrature points and nonzero quadrature weights, is equivalent to a collocation scheme, and so its convergence properties are fully determined by the previous collocation analysis and they depend only on the quadrature point location (in particular, they are completely independent of the accuracy of the quadrature rule). We also give a complete analysis for QDG with more than m + 1 quadrature points when the degree of precision (d.o.p.) is at least 2 m + 1. The behaviour (but not the approximation) is the same as that for a DG scheme when the d.o.p. is at least 2 m + 2. Numerical test results confirm that the theoretical convergence rates are optimal.

Description: The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. A discretization of this system is proposed in Part 1 of this work, where the displacements are approximated by standard finite elements and the action by a local postprocessing, which admits an equivalent mixed reformulation. Here we perform the a posteriori analysis of this discretization and prove optimal error estimates. Then we present numerical experiments that confirm the efficiency of the error indicators.

Description: In this paper we design high-order accurate and stable finite difference schemes for the initial–boundary–value problem associated with the magnetic induction equation with resistivity. We use summation-by-parts finite difference operators to approximate spatial derivatives and a simultaneous approximation term technique for implementing boundary conditions. The resulting schemes are shown to be energy stable. Various numerical experiments demonstrating both the stability and the high order of accuracy of the schemes are presented.

Description: Time-stepping procedures for the solution of evolution equations can be performed on parallel architecture by parallelizing the space computation at each time step. This, however, requires heavy communication between processors and becomes inefficient when many time steps are to be computed and many processors are available. In such cases parallelization in time is advantageous. In this paper we present a method for parallelization in time of linear multistep discretizations of linear evolution problems; we consider a model parabolic problem and a model hyperbolic problem and their, respectively, A ()-stable and A -stable linear multistep discretizations. The method consists of a discrete decoupling procedure, whereby N +1 decoupled Helmholtz problems with complex frequencies are obtained; N being the number of time steps computed in parallel. The usefulness of the method rests on our ability to solve these Helmholtz problems efficiently. We discuss the theory and give numerical examples for multigrid preconditioned iterative solvers of relevant complex frequency Helmholtz problems. The parallel approach can easily be combined with a time-stepping procedure, thereby obtaining a block time-stepping method where each block of steps is computed in parallel. In this way we are able to optimize the algorithm with respect to the number of processors available, the difficulty of solving the Helmholtz problems and the possibility of both time and space adaptivity. Extensions to other linear evolution problems and to Runge–Kutta time discretization are briefly mentioned.

Description: The Easter Seamount Chain and Nazca Ridge are two of the most conspicuous volcanic features on the Nazca plate. Many questions about their nature and origin have remained unresolved because of a lack of geochronological and geochemical data for large portions of both chains. New 40 Ar– 39 Ar incremental heating age determinations for dredged rocks from volcanoes east of Salas y Gomez Island show that, with very few exceptions, ages increase steadily to the east from 1·4 to 30 Ma, confirming that the two chains are parts of the same hotspot trail and indicating a hotspot location near Salas y Gomez rather than beneath Easter Island some 400 km farther west. Most of the volcanoes appear to have been erupted onto seafloor that was 5–13 Myr old, and no systematic variation in seafloor age at the time of seamount formation is apparent. At about 23 Ma, the formation of the Nazca Ridge ceased and that of the Easter Seamount Chain began, corresponding to a change in the direction of motion of the Nazca plate. Most of the studied rocks are moderately alkalic to transitional basalts. Their geochemical characteristics suggest that they represent relatively small mean amounts of partial melting initiating in garnet-bearing mantle and ending in the spinel facies. Nd–Sr–Pb isotopic compositions are within the range of values previously observed for volcanoes of the Easter Seamount Chain, west of Easter Island; moreover, most of our data cluster in a rather small part of this range [e.g. Nd (t) is between +6·0 and +4·0]. The results indicate that the mantle source has consisted of the same two principal components, a C/FOZO-type component and a high- Nd , incompatible-element-depleted Pacific mid-ocean ridge basalt-source-type component, since at least 30 Ma. The lack of any geochemical gradient along the chain east of Salas y Gomez implies that no systematic change over time has occurred in the proportions of these end-members.

Description: Silicic magma systems are of great scientific interest and societal importance owing to their role in the evolution of the crust and the hazards posed by volcanic eruptions. MELTS is a powerful and widely used tool to study the evolution of magmatic systems over a wide spectrum of compositions and conditions. However, the current calibration of MELTS fails to correctly predict the position of the quartz + feldspar saturation surface in temperature, pressure and composition space, making it unsuitable to study silicic systems. We create a modified calibration of MELTS optimized for silicic systems, dubbed rhyolite-MELTS, using early erupted Bishop pumice as a reference. Small adjustments to the calorimetrically determined enthalpy of formation of quartz and of the potassium end-member of alkali feldspar in the MELTS calibration lead to much improved predictions of the quartz + feldspar saturation surface as a function of pressure. Application of rhyolite-MELTS to the Highland Range Volcanic Sequence (Nevada), the Peach Spring Tuff (Arizona–Nevada–California), and the late-erupted Bishop Tuff (California), using compositions that vary from trachydacite to high-silica rhyolite, shows that the calibration is appropriate for a variety of fluid-bearing silicic systems. Some key observations include the following. (1) The simulated evolutionary paths are consistent with petrographic observations and glass compositions; further work is needed to compare predicted and observed mineral compositions. (2) The nearly invariant nature of silicic magmas is well captured by rhyolite-MELTS; unusual behavior is observed after extensive pseudo-invariant crystallization, suggesting that the new calibration works best for relatively small (i.e. 〈50 wt %) crystallization intervals, comparable with what is observed in volcanic rocks. (3) Our success with rhyolite-MELTS shows that water-bearing systems in which hydrous phases do not play a critical role can be appropriately handled; simulations are sensitive to initial water concentration, and although only a pure-H 2 O fluid is modeled, suitable amounts of water can be added or subtracted to mimic the effect of CO 2 in fluid solubility. Our continuing work on natural systems shows that rhyolite-MELTS is very useful in constraining crystallization conditions, and is particularly well suited to explore the eruptive potential of silicic magmas. We show that constraints placed by rhyolite-MELTS simulations using late-erupted Bishop Tuff whole-rock and melt inclusion compositions are inconsistent with a vertically stratified magma body.

Description: Jeju is a volcanic field that has erupted from around 1·8 Myr to c. 1 kyr ago. Activity began with dispersed, basaltic, monogenetic, phreatomagmatic eruptions. Continuing monogenetic volcanism was later joined by more voluminous lava effusion events building a central composite shield. Samples from older (〉0·7 Ma) and younger (〈0·2 Ma) monogenetic centres were analysed for their whole-rock major element, trace element and Sr–Nd–Pb isotopic compositions. Pyroclastic products from the monogenetic centres are dominantly alkali basalt to trachybasalt, whereas the more voluminous lava flows and domes of the central edifice consist of subalkali basalt and alkali basalt to trachyte. Lavas from the Early Pleistocene monogenetic centres are depleted in MgO, Cr and Ni, reflecting considerable olivine fractionation. By contrast, Late Pleistocene–Holocene monogenetic centre magmas fractionated clinopyroxene + olivine at deeper levels. Isotopic compositions show little variation across the suite; however, the Late Pleistocene–Holocene monogenetic centres have generally lower 87 Sr/ 86 Sr and 208 Pb/ 204 Pb and higher 143 Nd/ 144 Nd than the older centres and subalkali lavas. Major and trace element and isotope data suggest a common, shallower source for the high-Al alkali and subalkali lavas, in contrast to a deeper source for the low-Al alkali magmas. We propose that mantle melting was initiated under partially hydrous conditions at a pressure of near 2·5 GPa, followed by drier conditions and extension of the melting zone to 3–3·5 GPa, with a concomitant increase in the volume of melt derived from the shallower part of the system to produce subalkaline magmas. Increasing melt production at shallow depths may be related to accelerated heat transfer resulting from deepening of the melting zone, or increased mantle upwelling. Mantle lenses were uplifted, probably lubricated by shear zones created during the opening of the Sea of Japan c. 15 Myr ago, and reactivated during rotation of the Philippine Sea plate direction of subduction at around 2 Ma. This is the first hypothesized link between subduction processes and intraplate volcanism at Jeju.

Description: Partial melting of deep continental crust may occur during either prograde heating or decompression. Although the effect of temperature on crustal melting has been widely investigated, few experimental studies have addressed the question of the influence of pressure on crustal anatexis. To understand the influence of decreasing pressure on partial melting processes, the thermodynamic approach of isochemical phase diagrams has been applied to garnet–K-feldspar–kyanite–sillimanite anatectic gneisses (Barun Gneiss) from the Higher Himalayan Crystallines (HHC) of eastern Nepal. The main melt-producing reactions, the amount of melt produced during heating vs decompression, and the effects of melt loss on the mineral assemblages and compositions have been investigated along four ideal P–T trajectories, dominated by either heating or decompression. Based on these results, the observed microstructures and mineral compositions of the Barun Gneiss have been interpreted in terms of melt-producing vs melt-consuming reactions (e.g. growth of peritectic garnet with preserved ‘nanogranite’ inclusions vs microstructures related to back-reactions between solids and melt), and used to derive the metamorphic evolution of the studied samples. The P–T pseudosection modelling predicts that at least 15–20 vol. % of melt was produced at peak P–T conditions through dehydration melting of both muscovite and biotite, and that melt production was mainly triggered by heating, with or without the combined effect of decompression. The preserved granulitic peak metamorphic assemblage, however, is consistent with a significant loss of most of this melt. The P–T evolution inferred for samples from different, strategically located, structural levels of the Barun Gneiss is consistent with the expectations of a ‘channel flow’ model, including: (1) the clockwise shape of the P–T paths; (2) the estimated P at peak T (new data: 10–8 kbar at 800°C; model: 13–7 kbar at 800°C); (3) the decreasing P structurally upward, which defines a ‘normal’ metamorphic sequence, in contrast to the inverted metamorphic sequence occurring in the lowermost Main Central Thrust Zone; (4) the nearly isothermal exhumation of the structurally lowest sample, reflecting the progressive exhumation of rocks that have been entrained in the deep, high-T region of the channel, versus the nearly isobaric heating of the structurally uppermost sample, reflecting the evolution of those rocks that flowed outwards with the underlying channel.

Description: P–T–X(composition) pseudosections constructed for natural monzodioritic to peridotgabbroic rock compositions using T hermocalc in the model system Na 2 O–CaO–K 2 O–FeO–MgO–Al 2 O 3 –SiO 2 –H 2 O–TiO 2 –O (NCKFMASHTO) illustrate the dependence of granulite and eclogite assemblages on whole-rock composition at mid-crustal to upper-mantle conditions. Increasing ferric iron content results in a marked contraction of garnet–plagioclase assemblages, and an expansion of orthopyroxene, kyanite, quartz and ilmenite stability across the compositional range of monzodioritic–gabbroic protoliths, and the expansion of plagioclase stability up-pressure in gabbroic–peridotgabbroic compositions. Omphacite granulite defines a transitional stage between garnet granulite and eclogite in monzodioritic to gabbroic compositions. Silicate liquid compositions calculated for a monzodioritic protolith using the haplogranite melt model do not accurately reflect trends in comparable experimental data, and refinement is needed for its application to the modeling of intermediate and mafic equilibria involving more than a few per cent partial melt. Omphacite-bearing granulite mineral equilibria in dioritic protoliths are far less sensitive to changes in whole-rock oxidation state than gabbroic protoliths; a doubling of whole-rock oxygen content displaces the modeled granulite–eclogite transition in gabbroic assemblages by 0·4 GPa up-pressure. Results of low-H 2 O, high-O equilibria modeling best validate natural assemblages from Breaksea Sound, Fiordland, New Zealand, where co-facial monzodioritic granulite and peridotgabbroic eclogite formed at P 1·8 GPa and T 850°C.

Description: The range in An content of plagioclase in grain mounts of igneous cumulates provides a measure of diversity that is uniquely preserved in plagioclase because of its well-known refractory nature. To a first approximation such data provide, when calibrated, an estimate of the residual porosity or fraction of trapped liquid, in each specimen. The ensemble of specimens then provides a model for the stratigraphic variation of residual porosity. The raw data, however, include pre-cumulus zoning that can be isolated from in situ zoning by textural analysis in thin section. The baseline of residual porosity determinations was earlier determined for the Lower Zone of the Kiglapait intrusion from the content of excluded components in the solid rock compared with their content in the melt as calculated by summation and Rayleigh fractionation. The baseline equation was then used to calibrate the residual porosity obtained from the An range in grain mounts. This calibration is now extended to the remainder of the intrusion. The An range and the calculated residual porosity decrease to zero at 99% solidified (PCS) and then rise to the end of crystallization. The data suggest initial porosities smaller than 0·35. Allowing for pre-cumulus zoning, the data suggest a dominance of adcumulates in the intrusion and these impermeable barriers occupy 75% of the rocks in the Lower Zone. They occur at intervals of 1 to rarely 15 m and thereby restrict the likelihood of compaction over thick mushy zones. Variations in the Fo range of olivine are also observed in grain mounts and they follow those in plagioclase. However, they are in part due to subsolidus equilibration with Fe–Ti oxides and augite. The new calibration is successfully applied to the Skaergaard intrusion to supplement the published results from excluded components, with some interesting contradictions.

Description: High-Mg ultrapotassic volcanic rock occurrences of lamproitic affinity are exposed in southwestern Anatolia, mostly within the Menderes Massif. From north to south the lamproitic volcanism shows increasingly younger ages ranging from 20 to 4 Ma. Volcanism is contemporaneous with more voluminous shoshonitic, high-K calc-alkaline, and ultrapotassic magmatic activity in the Simav–Selendi, Usak, Kirka, Köroglu, Afyon and Isparta–Gölcük areas. The southward decrease in the age of the volcanism correlates with changes in geochemical composition, particularly a decrease in 87 Sr/ 86 Sr, 207 Pb/ 204 Pb, Zr/Nb and Th/Nb, and an increase in 143 Nd/ 144 Nd, 176 Hf/ 177 Hf, 206 Pb/ 204 Pb, 208 Pb/ 204 Pb and Ce/Pb, thus delineating a systematic change from orogenic (crust-like) to anorogenic (convecting mantle-like) signatures. Rare earth element compositions of clinopyroxene phenocrysts demonstrate an increasing role for residual garnet for locations in the central parts of the Menderes Massif, indicating a lithosphere thickness greater than 80 km. In contrast, K 2 O abundances remain nearly constant at around 7%, indicating buffering by phlogopite in the mantle source. Magma genesis in southwestern Anatolia is controlled by post-collisional extensional events initiated after major lithospheric thickening. Geochemical constraints suggest that the mantle source experienced two main geodynamic stages. The first stage caused ultradepletion of the mantle and subsequent metasomatic enrichment, which allowed coupling of the geochemical signatures of ultradepleted harzburgite with those of crust-derived sediments. This happened during the final closure stages of the southern Neotethys Ocean and the accretion of forearc oceanic lithosphere (island-arc type), as shallowly subducted material to the Anatolian lithosphere. The second stage is post-collisional, and is related to the collapse of the orogenic belt and the development of extension-related horst and graben structures. This stage is concurrent with the initiation of a thermal anomaly originating from a gap, identified by seismic tomography, in the subducted slab under western Anatolia. We propose that the lithospheric mantle underwent intense ‘asthenospherization’ owing to lithosphere–asthenosphere interaction, caused by the southward expansion of this gap during slab roll-back. The geochemical resemblance of the lamproites to more voluminous, contemporaneous shoshonitic magmas implies their derivation from a heterogeneous mantle source that had been affected by similar processes. These mantle processes may be closely associated with the major episode of uplift in the Menderes Massif.

Description: We identify olivine grains with compositions up to Fo 99 ·8 , which are found in multiple primitive basaltic lava flows from a monogenetic volcano in the Big Pine Volcanic Field, California, USA. In this study, we show that the forsterite in these basalts formed by subsolidus recrystallization in a high- f O 2 environment. Olivine compositions are bimodal, with flows having either all normal compositions (Fo 74 ·9 – 94 ·4 ) or highly forsteritic (Fo 97 ·2 – 99 ·8 ) compositions. In many grains, the subhedral forsteritic olivine has a hematite and clinopyroxene rim, and internal parallel-oriented planes of hematite, clinopyroxene and orthopyroxene. Results of isotopic, chemical, crystallographic, petrographic and mineralogical analyses show that the forsterite formed through subsolidus oxidation of olivine phenocrysts. The forsteritic olivines generally occur in the thinner flows. We infer that a rapidly emplaced sequence of thin, vesicular, spatter-fed flows allowed the original olivine phenocrysts to become repeatedly reheated while exposed to air. Our study required sampling each flow, so it was difficult to avoid the altered portions of the thinner flows. Other studies would tend to avoid such flows, which may account for why such forsteritic olivines have not been more widely recognized.

Description: Albitization is a common process during which hydrothermal fluids convert plagioclase and/or K-feldspar into nearly pure albite; however, its specific mechanism in granitoids is not well understood. The c . 1700 Ma A-type metaluminous ferroan granites in the Khetri complex of Rajasthan, NW India, have been albitized to a large extent by two metasomatic fronts, an initial transformation of oligoclase to nearly pure albite and a subsequent replacement of microcline by albite, with sharp contacts between the microcline-bearing and microcline-free zones. Albitization has bleached the original pinkish grey granite and turned it white. The mineralogical changes include transformation of oligoclase (~An 12 ) and microcline (~Or 95 ) to almost pure albite (~An 0 · 5 – 2 ), amphibole from potassian ferropargasite ( X Fe 0·84–0·86) to potassic hastingsite ( X Fe 0·88–0·97) and actinolite ( X Fe 0·32–0·67), and biotite from annite ( X Fe 0·71–0·74) to annite ( X Fe 0·90–0·91). Whole-rock isocon diagrams show that, during albitization, the granites experienced major hydration, slight gain in Si and major gain in Na, whereas K, Mg, Fe and Ca were lost along with Rb, Ba, Sr, Zn, light rare earth elements and U. Whole-rock Sm–Nd isotope data plot on an apparent isochron of 1419 ± 98 Ma and reveal significant disturbance and at least partial resetting of the intrusion age. Severe scatter in the whole-rock Rb–Sr isochron plot reflects the extreme Rb loss in the completely albitized samples, effectively freezing 87 Sr/ 86 Sr ratios in the albite granites at very high values (0·725–0·735). This indicates either infiltration of highly radiogenic Sr from the country rock or, more likely, radiogenic ingrowth during a considerable time lag (estimated to be at least 300 Myr) between original intrusion and albitization. The albitization took place at ~350–400°C. It was caused by the infiltration of an ascending hydrothermal fluid that had acquired high Na/K and Na/Ca ratios during migration through metamorphic rocks at even lower temperatures in the periphery of the plutons. Oxygen isotope ratios increase from 18 O = 7 in the original granite to values of 9–10 in completely albitized samples, suggesting that the fluid had equilibrated with surrounding metamorphosed crust. A metasomatic model, using chromatographic theory of fluid infiltration, explains the process for generating the observed zonation in terms of a leading metasomatic front where oligoclase of the original granite is converted to albite, and a second, trailing front where microcline is also converted to albite. The temperature gradients driving the fluid infiltration may have been produced by the high heat production of the granites themselves. The confinement of the albitized granites along the NE–SW-trending Khetri lineament and the pervasive nature of the albitization suggest that the albitizing fluids possibly originated during reactivation of the lineament. More generally, steady-state temperature gradients induced by the high internal heat production of A-type granites may provide the driving force for similar metasomatic and ore-forming processes in other highly enriched granitoid bodies.

Description: A rare occurrence of a chill sequence in drill core from the eastern Bushveld Complex has been discovered at the base of a thick succession of ultramafic rocks that forms part of the Lower Zone. The lowest 10 m of the section preserves a variety of rock types including a true chill against quartzite floor rock, crystalline quench-textured and spinifex-textured rocks as well as high-temperature olivine and orthopyroxene cumulates. It represents the first stage of magma emplacement into the Bushveld chamber and gives an insight into the nature of the parental magmas to the Bushveld, the processes that took place at that early stage, and how rocks with a high original interstitial liquid content give way to more normal cumulates higher in the section. The chill sequence also provides insight into the variety of rock types that are encountered in marginal sills that are regarded as representing early stage magmas intruded as the chamber developed but that are incompletely understood because of inadequate field exposures. Olivine compositions (up to Mg# 0·912) in a pyroxene dunite in this section are the highest recorded for the Bushveld Complex and cores of associated orthopyroxene have Mg# 0·93. Zoned orthopyroxenes in the quench- and spinifex-textured units range from Mg# 0·91 to 0·72 and preserve core compositions close to the original liquidus. Small single chromite crystals have Cr/(Cr + Al) of 0·85 and Cr/Fe(Total) of three; these are also the most primitive compositions found to date in the Bushveld Complex. The chill and quench zones represent the earliest magmas to be emplaced in the Bushveld Complex; however, these are relatively evolved and similar to the B1 liquid, long assumed to be the most primitive magma that gave rise to the Lower Zone. Major and trace element geochemistry and mineral compositions show that the true parental magmas to the Lower Zone were of komatiite composition, with the most primitive containing 19% MgO. The liquid compositions have a strong crustal signature but are also enriched in K, Rb, Pb and especially Cs and Cl. The olivines in this section are highly Ni enriched, consistent with a component of mantle pyroxenite derived from recycled ocean crust in the source of the parental magmas. It is suggested that this source, combined with PGE-fertile subcontinental lithospheric mantle, was melted within a rising mantle plume and the resultant melts then interacted with the basement rocks of the Kapvaal Craton.

Description: Tofua volcano is situated midway along the Tonga oceanic arc and has undergone two phases of ignimbrite-forming activity. The eruptive products are almost entirely basaltic andesites (52·5–57 wt % SiO 2 ) with the exception of a volumetrically minor pre-caldera dacite. The suite displays a strong tholeiitic trend with K 2 O 〈1 wt %. Phenocryst assemblages typically comprise plagioclase + clinopyroxene ± orthopyroxene with microlites of Ti-magnetite. Olivine (Fo 83 – 88 ) is rare and believed to be dominantly antecrystic. An increase in the extent and frequency of reverse zoning in phenocrysts, sieve-textured plagioclase and the occurrence of antecrystic phases in post-caldera lavas record a shift to dynamic conditions, allowing the interaction of magma batches that were previously distinct. Pyroxene thermobarometry suggests crystallization at 950–1200°C and 0·8–1·8 kbar. Volatile measurements of glassy melt inclusions indicate a maximum H 2 O content of 4·16 wt % H 2 O, and CO 2 –H 2 O saturation curves indicate that crystallization occurred at two levels, at depths of 4–5·5 km and 1·5–2·5 km. Major and trace element models suggest that the compositions of the majority of the samples represent a differentiation trend whereby the dacite was produced by 65% fractional crystallization of the most primitive basaltic andesite. Trace element models suggest that the sub-arc mantle source is the residuum of depleted Indian mid-ocean ridge basalt mantle (IDMM-1% melt), whereas radiogenic isotope data imply addition of 0·2% average Tongan sediment melt and a fluid component derived from the subducted altered Pacific oceanic crust. A horizontal array on the U–Th equiline diagram and Ra excesses of up to 500% suggest fluid addition to the mantle wedge within the last few thousand years. Time-integrated ( 226 Ra/ 230 Th) vs Sr/Th and Ba/Th fractionation models imply differentiation timescales of up to 4500 years for the dacitic magma compositions at Tofua.

Description: Fonualei is unusual amongst subaerial volcanoes in the Tonga arc because it has erupted dacitic vesicular lavas, tuffs and phreomagmatic deposits for the last 165 years. The total volume of dacite may approach 5 km 3 and overlies basal basaltic andesite and andesite lavas that are constrained to be less than a few millennia in age. All of the products are crystal-poor and formed from relatively low-viscosity magmas inferred to have had temperatures of 1100–1000°C, 2–4 wt % H 2 O and oxygen fugacities 1–2 log units above the quartz–fayalite–magnetite buffer. Major and trace element data, along with Sr–Nd–Pb and U–Th–Ra isotope data, are used to assess competing models for the origin of the dacites. Positive correlations between Sc and Zr and Sr rule out evolution of the within-dacite compositional array by closed-system crystal fractionation of a single magma batch. An origin by partial melting of lower crustal amphibolites cannot reproduce these data trends or, arguably, any of the dacites either. Instead, we develop a model in which the dacites reflect mixing between two dacitic magmas, each the product of fractional crystallization of basaltic andesite magmas formed by different degrees of partial melting. Mixing was efficient because the two magmas had similar temperatures and viscosities. This is inferred to have occurred at shallow (2–6 km) depths beneath the volcano. U–Th–Ra disequilibria in the basaltic andesite and andesite indicate that the parental magmas had fluids added to their mantle source regions less than 8 kyr ago and that fractionation to the dacitic compositions took less than a few millennia. The 165 year eruption period for the dacites implies that mixing occurred on a similar timescale, possibly during ascent in conduits. The composition of the dacites renders them unsuitable candidates as contributors to average continental crust.

Description: We performed a thermodynamic and experimental study to investigate the fate of pyroxenite-derived melts during their migration through the peridotitic mantle. We used a simplified model of interaction in which peridotite is impregnated by and then equilibrated with a finite amount of pyroxenite-derived liquid. We considered two pyroxenite compositions and three contexts of pyroxenitic melt impregnation: (1) in a subsolidus lithospheric mantle; (2) beneath a mid-ocean ridge (MOR) in a subsolidus asthenospheric mantle at high pressure; (3) beneath a MOR in a partially molten asthenospheric mantle. Calculations were performed with pMELTS at constant pressure and temperature with a melt–rock ratio varying in the range 0–1. Concurrently, a series of impregnation experiments was performed at 1 and 1·5 GPa to reproduce the final stages of the calculations where the melt–rock ratio is 1. Incoming melt and host-rocks react differently according to the melt composition and the physical state of the surrounding mantle. Whereas clinopyroxene (Cpx) is systematically a reaction product, the role of olivine (Ol) and orthopyroxene (Opx) depends on the incoming melt silica activity a SiO 2 : if it is lower than the silica activity $${\hbox{ a }}_{Si{O}_{2}}^{0}$$ of a melt saturated in Ol and Opx at the same pressure P and temperature T , Opx is dissolved and Ol precipitates, and conversely if a SiO 2 〉 $${\hbox{ a }}_{Si{O}_{2}}^{0}$$ . Such contrasted reactions between pyroxenitic melts and peridotitic mantle may generate a large range of new lithological heterogeneities (wehrlite, websterite, clinopyroxenite) in the upper mantle. Also, our study shows that the ability of pyroxenite-derived melts to migrate through the mantle depends on the melting degree of the surrounding peridotite. The reaction of these melts with a subsolidus mantle results in strong melt consumption (40–100%) and substantial Cpx production (with some spinel or garnet, depending on P ). This is expected to drastically decrease the system permeability and the capacity of pyroxenite-derived melts to infiltrate neighbouring rocks. In contrast, melt migration to the surface should be possible if the surrounding mantle is partially melted; although liquid reactivity varies with composition, melt consumption is restricted to less than 20%. Hence, magma–rock interactions can have a significant impact on the dynamics of melting and magma migration and should not be neglected when modelling the partial melting of heterogeneous mantle.

Description: Low-grade metabasites and hydrothermally altered ultramafic rocks form most of the Paleoarchean Barberton Greenstone Belt (BGB) of South Africa. However, P – T constraints are sparse and the nature of the greenschist-facies metamorphism is poorly characterized. This study provides new P – T estimates and descriptions of the petrological characteristics of altered mafic–ultramafic rocks across the Kromberg type-section, Onverwacht Group, BGB, from both surface samples and fresh drill core. Application of a chlorite thermodynamic multi-equilibrium calculation and pseudosection modelling, in conjunction with petrographic observations, indicates a wide range in metamorphic conditions from sub-greenschist to the uppermost greenschist facies across the type-section. A central fuchsite-bearing zone containing strong mylonitic fabrics, referred to as the Kromberg Section Mylonites (KSM), records at least two metamorphic events: a high- T , low- P (420 ± 30°C, 〈3 kbar) metamorphism, and a lower-T event ( T = 240–350°C, P = 2·9 ± 0·15 kbar) related to retrograde metamorphism associated with extensional quartz–carbonate veins. Pillow lava metabasalts directly beneath the KSM record the same HT–LP-type metamorphic conditions, whereas metabasalts 90 m and 125 m above the KSM record temperatures of 250–340°C. Lower in the Kromberg section, metamorphic conditions vary from 215–321°C (at 2·9 kbar, mid to lower parts) to very low-grade conditions of 140–209°C in the lowermost Kromberg. Thus, an inverted metamorphic field gradient is documented beneath the KSM. Petrological analysis of the fuchsite-bearing rocks of the KSM indicates that they contain listvenite, a hydrothermal alteration product typically found in mafic–ultramafic tectonic mélange zones of ophiolite terranes. Together with the new metamorphic P – T constraints this means that these rocks are not a product of Archean atmospheric weathering. Rather, the inverted metamorphic field gradient in the Kromberg type-section suggests that the KSM represents a Paleoarchean thrust zone. Cr-spinel compositions in the ultramafic protolith to the KSM listvenites suggest a supra-subduction zone setting. A new geodynamic model is therefore proposed in which the mafic–ultramafic oceanic rocks of the Kromberg type-section were obducted as a thrust nappe pile in a regional transpressional tectonic regime between 3227 and 3230 Ma.

Description: Thickening of arc lithosphere influences the extent of magmatic differentiation and is thereby important for the evolution of juvenile arcs into mature continental crust. Here, we use mantle xenoliths from the late Mesozoic Sierra Nevada continental arc in California (USA) to constrain the pressure, temperature, and compositional evolution of the deep lithosphere beneath a mature arc. These xenoliths consist of spinel peridotites and garnet-bearing spinel peridotites. The former are characterized by coarse-grained protogranular textures having bulk compositions indicative of high-degree melting. The latter are characterized by porphyroclastic textures, garnet coronas around spinels, garnet exsolution lamellae in pyroxenes, and pyroxenes with high-Al cores and low-Al rims. The garnet-bearing spinel peridotites range from depleted to fertile compositions, but the high Cr-numbers [molar Cr/(Cr + Al)] of spinel cores reflect high-degree melting. These observations suggest that the protoliths of the garnet-bearing spinel peridotites were melt-depleted spinel peridotites. Constraints from geothermobarometry and bulk compositions coupled with mantle melting models suggest that the protoliths underwent shallow melt depletion (1–2 GPa, 1300–1400°C), followed by compression, cooling, and final equilibration within the garnet stability field (~3 GPa, 〈800°C). The deepest equilibrated samples are the most refertilized, suggesting that refertilization occurred during compression. We interpret this P – T –composition path to reflect progressive thickening of the Sierran arc lithosphere perhaps as a result of magmatic inflation or tectonic thickening. We hypothesize that newly formed arc lithospheric mantle thickens enough to pinch out the asthenospheric wedge, juxtaposing Sierran arc lithosphere against the subducting oceanic plate. This could have terminated arc magmatism and initiated cooling of the deep Sierran lithosphere.

Description: In the Okoppe area of North Hokkaido, Japan, the eruption of adakitic and calc-alkaline dacites was followed by high-Mg andesite (HMA) and calc-alkaline dacite during the Middle Miocene (12–10 Ma). Adakitic dacite is characterized by high Sr/Y and low Y, with Sr and Nd isotopic compositions identical to those of mid-ocean ridge basalt. It has higher MgO contents than adakites generated by experimental melting of metabasalt and amphibolite, and higher Ni and Cr contents than either Archean trondhjemite–tonalite–granodiorite or Early Cretaceous adakitic granites, which are considered to represent partial melts of subducted oceanic crust. This provides compelling evidence that adakitic dacite magma from Okoppe resulted from interaction of a melt derived from subducted oceanic basaltic crust and the overlying mantle wedge peridotite, with little modification to the adakitic melt signature and Sr and Nd isotopic values. The compositional variations in the Toyono adakitic dacite and associated calc-alkaline dacite probably resulted from mixing of the reacted magma and an evolved silicic dacite magma formed by fractional crystallization of the reacted magma. A disequilibrium phenocryst assemblage in the HMA may result from mixing of boninite and silicic andesite that resulted from crustal melting. Calc-alkaline dacites associated with the HMA were derived by fractional crystallization of silicic andesite and assimilation of crust with an enriched Sr isotopic signature. The most likely tectono-magmatic model for the production of adakitic dacite and HMA involves upwelling of hot asthenosphere into the subcontinental lithosphere beneath North Hokkaido and the back-arc side of the NE Japan arc, coincident with the spreading of the Kurile back-arc basin and Japan Sea back-arc basin. This resulted in a high geothermal gradient in the mantle wedge beneath North Hokkaido. The subsequent melting of a limited part of the cool oceanic crust subducting beneath Hokkaido produced adakitic magmas, which interacted with the overlying mantle wedge peridotite. These magmas subsequently reacted with an evolved calc-alkaline melt en route to the surface. Boninitic magma derived from the ascending hot asthenosphere in part reacted with crust-derived silicic andesitic magma, undergoing simultaneous fractional crystallization.

Description: Experiments at 20–97 GPa and 2000 K in the system CaO–MgO–TiO 2 –SiO 2 constrain phase relations involving Mg-rich and Ca-rich perovskite solid solutions at conditions relevant to the Earth's deep Transition Zone and lower mantle. Bulk compositions were investigated with molar Ti/(Ti + Si) up to 0·5 within the quasi-ternary ‘perovskite plane’, which is defined by a reciprocal solution among the components MgSiO 3 , MgTiO 3 , CaSiO 3 , and CaTiO 3 . Multi-anvil experiments at 20 GPa and 2000 K on bulk compositions within the plane produce akimotoite coexisting with Ca-perovskites that lie close to the CaSiO 3 –CaTiO 3 join. Higher-pressure experiments using a laser-heated diamond anvil cell constrain the position of a two-perovskite field that extends into the perovskite plane from the solvus along the MgSiO 3 –CaSiO 3 binary join, where limited mutual solubility exists between MgSiO 3 and CaSiO 3 perovskites. On the join MgSiO 3 –MgTiO 3 , MgTiO 3 solubility in MgSiO 3 perovskite increases with pressure, with MgSi 0 · 8 Ti 0 · 2 O 3 perovskite stable at ~50 GPa. Limited reciprocal solution at ~25 GPa results in an expansive two-perovskite field that occupies much of the Si-rich portion of the perovskite plane. Solution of Ti into Mg-rich and Ca-rich perovskites enhances the solubility of reciprocal Ca and Mg components, respectively. Increase in pressure promotes reciprocal solution, and the two-phase field collapses rapidly with pressure toward the MgSiO 3 –CaSiO 3 join. We find that a single-phase, orthorhombic perovskite with near equimolar Ca and Mg is stable in a composition with Ti/(Ti + Si) of only 0·05 at 97 GPa, requiring that by this pressure the two-phase field occupies a small area close to the MgSiO 3 –CaSiO 3 join. On the basis of experiments at~1500 K, temperature has only a mild effect on the position of the Ca-rich limb of the solvus. Ca(Ti,Si)O 3 mineral inclusions in deep sublithospheric diamonds could not have formed in equilibrium with Mg-perovskite owing to their virtual lack of MgSiO 3 component at pressures of Mg-perovskite stability, but may have equilibrated with Transition Zone MgSiO 3 -rich phases at lower pressures; this observation can be extended generally to near-endmember CaSiO 3 inclusions. On an iron-free basis, the average bulk compositions of clinopyroxene–ilmenite and orthopyroxene–ilmenite megacrysts from kimberlites plot in single-perovskite fields at pressures greater than about 45 and 65 GPa, respectively, when projected onto the perovskite plane. We predict that the effect of iron will not be large, and estimate that single-phase perovskites may form at somewhat lower pressures than in the iron-free system. Thus, the origin of pyroxene–ilmenite megacrysts from single-phase perovskite solutions in the lower mantle is plausible on the basis of phase relations, although a lower pressure magmatic origin appears more likely. Deeply subducted Ti-rich lithologies such as ocean-island basalt will crystallize a single perovskite rather than a two-perovskite assemblage beginning at pressures of ~80 GPa. Normal mid-ocean ridge basalt and primitive mantle peridotite are expected to remain within a two-phase perovskite field until Mg-perovskite transforms to post-perovskite.

Description: The Bishop Tuff is a giant silicic ignimbrite erupted at 0·76 Ma in eastern California, USA. Five pumice clasts from the late-erupted Bishop Tuff (Aeolian Buttes) were studied in an effort to better understand the pre- and syn-eruptive history of the Bishop magma body and place constraints on the timescales of its existence. This study complements and expands on a previous study that focused on early-erupted Bishop Tuff pumice clasts. Bulk densities of pumice clasts were measured using an immersion method, and phenocryst crystal contents were determined using a sieving and winnowing procedure. X-ray tomography was used to obtain qualitative and quantitative textural information, particularly crystal size distributions (CSDs). We have determined CSDs for crystals ranging in size from ~10 to ~1000 µm for three groups of mineral phases: magnetite (±ilmenite), pyroxene + biotite, quartz + feldspar. Similar to early-erupted pumice, late-erupted pumice bulk density and crystal contents are positively correlated, and comparison of crystal fraction vs size trends suggests that the proportion of large crystals is the primary control on crystallinity. Porosity is negatively correlated with crystal content, which is difficult to reconcile with closed-system crystallization. Magnetite and pyroxene + biotite size distributions are fractal in nature, often attributed to fragmentation; however, crystals are mostly whole and euhedral, such that an alternative mechanism is necessary to explain these distributions. Quartz + feldspar size distributions are kinked, with a shallow-sloped log–linear section describing large crystals (〉140 µm) and a steep-sloped log–linear section describing small crystals (〈140 µm). We interpret these two crystal populations as resulting from a shift in crystallization regime. We suggest that the shallow-sloped section describes a pre-eruptive quartz + feldspar growth-dominated regime, whereas the steep-sloped section represents a population that grew during a nucleation-dominated regime that began as a result of decompression at the onset of eruption. Timescales of quartz growth calculated from the slopes of these two segments of the size distributions indicate that the pre-eruptive crystal population grew on timescales on the order of millennia and may describe the timescale of crystallization of the Bishop magma body. The syn-eruptive population gives timescales of 〈1–2 years (but possibly much less) and probably marks the onset of eruptive decompression.

Description: We present the results of the first systematic study of melt compositions at Pantelleria, based on both melt inclusions and matrix glasses in pantellerites from 10 eruptions during the last eruptive cycle (〈45 kyr). We present major and trace element compositions, as well as data on the volatiles sulphur (S), fluorine (F), chlorine (Cl), water (H 2 O), carbon dioxide (CO 2 ) and lithium (Li) Rare earth element (REE) compositions were inverted using the program INVMEL to establish the melt fraction vs depth relationship in the Pantellerian mantle source region. Inversion indicates that melts are generated by ~1·7% melting of a light rare earth element (LREE)-enriched mantle source. The source lies principally within the spinel–garnet transition zone, which, on the basis of trace element ratios, shows some affinity to the source of North African magmatism. Major and trace element data indicate a gap in melt compositions at intermediate compositions, consistent with previously published whole-rock data. This gap rules out the possibility of explaining chemical variability in the Pantelleria lavas merely by changes in the crystal content of the magmas. Principal component analysis of major element glass compositions shows that the liquid line of descent for mafic melt compositions is controlled by clinopyroxene, plagioclase, magnetite and olivine crystallization. Alkali feldspar, clinopyroxene, ilmenite and olivine or aenigmatite crystallization controls the liquid line of descent for the silicic melt compositions, with aenigmatite broadly replacing olivine in the most evolved magmas. Trace element modelling indicates that 96% fractional crystallization is required to generate pantellerites from alkali basalts at Pantelleria (through trachytes, generated after 76% fractional crystallization). We have measured pantellerite volatile concentrations in melt inclusions and in matrix glasses from a variety of eruptions. Melt inclusions, on average, contain 350 ppm S, 3500 ppm F and 9000 ppm Cl. We have measured up to 4·9 wt % H 2 O and 150 ppm CO 2 in melt inclusions. Li–H 2 O systematics and Cl abundances in melt inclusions are consistent with partitioning of Li and Cl into a subcritical hydrosaline fluid at low pressures. The volatiles H 2 O and CO 2 are used to estimate melt equilibration pressures, which reach a maximum of 1·5 kbar. Temperatures of 800°C are calculated for the most evolved pantellerites, using published feldspar–melt geothermometers, and up to 870°C for the least evolved samples. Low melt viscosities are calculated for the range of pantellerite compositions observed and may account for rapid differentiation by crystal settling. Stable density stratification of the magma chamber is reflected in the eruption of generally progressively more fractionated compositions after the Green Tuff eruption during the last eruptive cycle. Some anomalies in this trend may be explained by variation in the relative rates of eruption vs fractionation. The density stratification is expected to be enhanced and further stabilized by the efficient migration of a fluid phase to the roof of the magma chamber. The sulphur data are used in combination with published experimental partitioning data for peralkaline rhyolites to estimate the sulphur yield to the atmosphere for a large pantelleritic eruption similar to the Green Tuff. This is expected to be markedly higher than for a similar-sized metaluminous rhyolitic or dacitic eruption, mainly owing to the higher bulk sulphur content, lower fluid–melt partition coefficients, and rapid differentiation and vapour phase segregation in the magma chamber.

Description: Magma mixing is a ubiquitous process in silicic magmatic systems. We present pre- and post-mixing time scales for a single magma mixing event associated with the 1846–1847 eruption of Volcán Quizapu. We compare and contrast 226 Ra– 230 Th crystal residence ages between the mixed lavas of the 1846–1847 effusive eruption and the largely homogeneous plinian dacite eruption of 1932. Both eruptions are dominated by the same dacite magma, whereas the earlier eruption contains evidence for a single episode of mafic recharge and mixing. The use of uranium-series mineral separate dating on a mixed magma requires deconvolution of different crystal populations. We present a four-component mixing model for the crystal separates to obtain dacite-derived plagioclase model ages as well as andesite-derived (recharge) plagioclase ages. Plagioclase model ages average ~3·6 ka for dacite-derived plagioclase from the mixed (1846–1847) eruption, ~1·4 kyr older than average plagioclase model ages of the 1932 eruption, suggesting that magma mixing and reheating led to some resorption of young crystal rims in the 1846–1847 magmatic episode and/or that some growth of plagioclase rims occurred during the repose time of 86 years. The andesite-derived plagioclase residence ages are younger than the corresponding dacite-derived plagioclase residence ages, but they are still of the order of thousands of years. In addition, we use Mg diffusion in plagioclase to constrain time scales of magma mixing. We show that the 1846–1847 eruption was triggered by large-volume mafic recharge magma that mixed and mingled with the resident dacite within probably days to weeks of the ensuing eruption.

Description: Crystal zoning as well as temperature and pressure estimates from phenocryst phase equilibria are used to constrain the architecture of the intermediate-sized magmatic system (some tens of km 3 ) of Volcán Quizapu, Chile, and to document the textural and compositional effects of magma mixing. In contrast to most arc magma systems, where multiple episodes of open-system behavior obscure the evidence of major magma chamber events (e.g. melt extraction, magma mixing), the Quizapu magma system shows limited petrographic complexity in two large historical eruptions (1846–1847 and 1932) that have contrasting eruptive styles. Quizapu magmas and peripheral mafic magmas exhibit a simple binary mixing relationship. At the mafic end, basaltic andesite to andesite recharge magmas complement the record from peripheral cones and show the same limited range of compositions. The silicic end-member composition is almost identical in both eruptions of Quizapu. The effusive 1846–1847 eruption records significant mixing between the mafic and silicic end-members, resulting in hybridized andesites and mingled dacites. These two compositionally simple eruptions at Volcán Quizapu present a rare opportunity to isolate particular aspects of magma evolution—formation of homogeneous dacite magma and late-stage magma mixing—from other magma chamber processes. Crystal zoning, trace element compositions, and crystal-size distributions provide evidence for spatial separation of the mafic and silicic magmas. Dacite-derived plagioclase phenocrysts (i.e. An 25 – 40 ) show a narrow range in composition and limited zonation, suggesting growth from a compositionally restricted melt. Dacite-derived amphibole phenocrysts show similar restricted compositions and furthermore constrain, together with more mafic amphibole phenocrysts, the architecture of the magmatic system at Volcán Quizapu to be compositionally and thermally zoned, in which an andesitic mush is overlain by a homogeneous dacitic magma that is the source for most of the 1846–1847 and 1932 erupted magmas. Dacite formation is best explained by mineral–melt separation (crystal fractionation) from an andesitic mush, which is inferred to have thermally and compositionally buffered the dacite magma thereby keeping it at relatively low crystallinity (〈30 vol. %). The dominant cause of compositional diversity is melt separation. Back-mixing of mush (i.e. crystals with signatures of growth both in the andesitic mush and in the dacite magma) into the overlying dacite magma is rarely observed. Recharge events that increase crystal and magma diversity in the dacite magma are limited to an episode of mafic recharge and mixing just prior to the 1846–1847 eruption, where evidence for magma mixing is present on all scales. Chamber-wide mixing was incomplete (mixing efficiency of ~0·53–0·85) as flow lobes vary significantly in composition along the proposed mixing array. Estimates of viscosity variations during the course of magma mixing suggest that mixing dynamics and the degree of magma interaction on all scales were established at the beginning of the recharge event.

Description: Decompression experiments have been conducted to simulate syn-eruptive crystallization in volcanic conduits, to infer magma ascent rates and conditions during dome-forming and Plinian eruptions of silicic arc volcanoes. The experiments were carried out on starting material with the composition of Mt. Pelée rhyolitic interstitial melt (76 wt % SiO 2 ) and consisted of three consecutive steps: hydration, decompression, and annealing. Hydration (saturated and undersaturated) was performed at 850°C and 200 MPa and was followed by isothermal decompression, either linearly or stepwise, to a final pressure, P f , of 30 or 5–10 MPa. Decompression rates range from 0·003 to 25 MPa min – 1 (decompression durations of 15 min to 40 days). Two samples were cooled by 25°C and 50°C during a 3 day step at P f . Subsequent to decompression, the samples were held for up to 15 days at P f . The experiments generated three types of crystals as products of pre-, syn-, and post-decompression crystallization. The experiments basically differ from previous studies in that they are specifically designed to discriminate crystal nucleation from growth and to evaluate the influence of pre-decompression crystals on the decompression-induced crystallization. The effects of pre-decompression crystals, decompression rate, undercooling ( P f ), and terminal cooling have been determined on plagioclase nucleation, growth, morphology, and composition. The main results suggest a positive correlation between decompression rate and the number density of plagioclase crystals nucleated at P f and highlight the effect of pre-decompression crystals in further decompression-induced crystallization. The relations between the decompression conditions and the plagioclase characteristics have been used to infer Mt. Pelée eruption dynamics, suggesting that (1) Plinian magmas ascend from the reservoir within less than 1 h (1–10 m s – 1 ), (2) dome and block-and-ash flow magmas ascend within more than 2–5 days, giving time for syn-decompression crystallization around pre-existent microlites, (3) dome magmas provide evidence for long stagnation and cooling at low pressure, and (4) surge magmas ascend without significant crystallization (within less than ~4 days) and massively nucleate plagioclase at very low pressure. The extent and violence of dome destruction may depend on the size or age of the dome, with large or old domes favouring mildly explosive block-and-ash flows, whereas small or young protodomes may generate highly explosive surges.

Description: Most published studies of olivine-hosted melt inclusions from mid-ocean ridges have been based on a single sample. Here we present a comprehensive melt inclusion study of major and trace elements from a single ocean ridge segment, the FAMOUS segment of the Mid-Atlantic Ridge. The melt inclusion dataset includes 312 olivine-hosted (Mg-number 85–92) melt inclusions from 14 samples distributed along the segment. This permits a more comprehensive assessment of the variability within melt inclusions from a single region, and of the relationship between melt inclusion and lava compositions. One recent question has been the extent to which melt inclusions truly preserve the original melt compositions, or instead are modified by late-stage processes occurring at shallow levels. In the FAMOUS inclusions, major elements have been affected by post-entrapment processes, but trace elements show no evidence of such processes, suggesting that diffusion coefficients for incompatible elements are small. Melt inclusions can be divided into three groups. (1) High-Mg inclusions are the most primitive and may potentially constrain the composition of the parental magmas that contribute to other melt inclusion and lava compositions. Although their trace element contents range from highly depleted to almost as enriched as the FAMOUS segment lavas, they are on average more depleted and the melts appear to be derived by greater extents of melting than the lavas. (2) Low-Al inclusions occur in the lower Mg-number olivines, and their major and trace element characteristics reflect mixing between high-Mg melt inclusion and lava compositions. (3) High-Al melt inclusions display Al 2 O 3 contents as high as 18·4 wt %, SiO 2 as low as 46·6 wt %, a strong depletion in the most incompatible elements and distinctively low middle to heavy rare earth element (MREE/HREE) ratios. The high Al 2 O 3 and low SiO 2 contents, as well as positive Sr anomalies in some of the high-Al melt inclusions, are best explained by assimilation of plagioclase-bearing cumulates. The trace element variability in the high-Mg melt inclusions is not consistent with a simple continuous melting column and requires pooling of near-fractional melts within the melting regime and a variable mantle source composition. Because the mean composition of these melt inclusions reflects greater extents of melting than the lavas, we propose that the melt inclusions come from the upper portions of the melting regime. Lavas, in contrast, sample the entire melting regime, including low-degree melts from the wings of the regime that are transported more directly to the surface along high-porosity channels. The high-Al, trace element ultra-depleted, low MREE/HREE melt inclusions derive from melting of a residual mantle source formed by previous melt extraction in the garnet stability field. There is a marked lack of correspondence between major and trace element variations in the melt inclusions. This may reflect a combination of processes, such as cumulate assimilation and re-equilibration of the magmas during ascent, which can reset major elements while having little effect on the trace element variations. The melt inclusions are not simply representative unpooled melts from the melting regime and they do not fully reflect the range of melt compositions contributing to the lavas. Their compositions reflect source heterogeneity as well as melting processes, and major and trace element indicators of depth of origin do not correspond. Combined comprehensive studies of lavas and melt inclusions have much more to reveal than studies based on either data source alone.

Description: The petrogenesis of calc-alkaline magmatism in the Famatinian arc is investigated in the central Sierra Valle Fértil, a major, lower to middle crustal section of the Early Ordovician active margin of West Gondwana. Large-scale field relationships show a gradual and continuous compositional variation of the plutonic sequence, ranging from olivine-bearing gabbronorites to hornblende- and biotite-bearing granodiorites. Distinctive lithostratigraphic units are, however, discernible as one compositional type of plutonic rock dominates over mappable areas. These results allow us to identify a continuous plutonic arc stratigraphy that progressively exposes shallower paleo-depths towards the east. At all the exposed levels, calc-alkaline plutonic rocks are volumetrically dominant, interrupted only by granulite-facies migmatites and leucogranites. The migmatites are interpreted to be refractory remnants of supracrustal sedimentary successions, whereas the peraluminous leucogranites have field relationships and chemical and isotopic compositions suggesting that they were produced via anatexis of metasedimentary packages. Mass-balance calculations predict that a parental gabbroic magma after progressive closed-system fractionation would crystallize about 80% of the original mass to yield a granodioritic daughter. Because the crystallizing mineral assemblage comprises hornblende and plagioclase, mass balance suggests a volume of residual amphibole-rich gabbroic rocks much larger than that observed, suggesting that differentiation is significantly driven by open-system processes. Indeed, the combination of field and petrographic observations with bulk-rock geochemistry and petrogenetic modeling demonstrates that most dioritic and tonalitic rocks are hybrids formed by either (1) bulk assimilation of metasedimentary materials into gabbroic magmas, or (2) multi-stage and complex interactions between gabbroic rocks and metasedimentary-derived leucogranitic melts. The source region of the granodioritic magmas is located at the transition zone between a tonalite-dominated intermediate unit and a granodiorite-dominated silicic unit. Typical granodiorites have a hornblende-bearing mineralogy, metaluminous chemical signature and isotopic compositions [ 87 Sr/ 86 Sr(T) = 0·7075–0·7100 and Nd (T) ~ –5·0] broadly overlapping those of the tonalites of the intermediate rock unit. These major compositional features of the granodiorites can be best explained if three end-member components contribute to their generation. As field observational data suggest, primitive gabbroic rocks, metaluminous intermediate magmas and anatectic leucogranitic melts mixed to produce the calc-alkaline granodiorites; however, the exact petrological process generating the granodioritic magmas is unclear because the mafic end-member may have been incorporated as mafic inclusions in the intermediate magmas or as syn-magmatic dikes, or both. The polygenetic nature of the intermediate to silicic plutonic rocks, along with the preponderance of parental gabbroic rocks at the inferred base of the plutonic column, suggests an upward growth of the intermediate to silicic crust that involved the complete reconstitution of the pre-existing crustal configuration. The main implication of this study is that intermediate and silicic plutonic rocks in the Valle Fértil section formed within a crustal column in which the mass transfer and heat input of mantle-derived magmas promoted fusion of fertile metasedimentary rocks and favored mixing of gabbroic or dioritic magmas with crustal granitic melts. Our results lend support to models asserting that the thermal and material budget of arc magmatism is primarily governed by the rate at which mafic magmas ascend from their mantle sources and intrude repeatedly into the crust.

Description: We have experimentally investigated a K-bearing altered mid-ocean ridge basalt (MORB) composition to which 10% CaCO 3 was added (GA1 + 10%cc), at temperatures of 1050–1400°C and pressures of 3·5–5·5 GPa. Experiments were conducted in piston-cylinder apparatus in Pt–Gr (Pt with inner graphite) and Au–Pd capsules. Sub-solidus assemblages for both sets of experiments contain clinopyroxene, garnet, carbonate, rutile, coesite and K-feldspar. Apatite was observed only in the Pt–Gr experiments. Melting behaviour in experiments using different capsule materials contrasted markedly. Experiments in Pt–Gr capsules showed the silicate solidus to be at temperatures less than 1100°C at 3·5 GPa and less than 1050°C at 4·5–5·0 GPa. These are similar (3·5 GPa) or lower (4·5–5·0 GPa) temperatures compared with the carbonate solidus (1075–1125°C at 3·5–5·0 GPa). Melts in the Pt–Gr runs evolve with increasing degree of melting from K-rich silicate melts at the lowest degree of melting to carbonate–silicate immiscible liquids and silicate–carbonate melts at intermediate degrees of melting, and finally to silicate melts at the highest degrees of melting. Experiments in Au–Pd capsules were performed only at 5·0 GPa. The carbonate solidus is between 1200 and 1225°C (at least 100°C higher than in the experiments in Pt–Gr capsules at the same pressure–temperature conditions). The first melts to be produced are carbonatitic and exhibit increasing SiO 2 content with increasing temperature. This contrast in melting behaviour is explained by the relatively rapid diffusion of H through the Pt–Gr capsules, resulting in formation of H 2 O, and thus dramatically depressing both the silicate and the carbonate solidi in the Pt–Gr experiments compared with those in the Au–Pd experiments. This presumably reflects the lower permeability of Au–Pd to H, resulting in a much lower H 2 O/CO 2 ratio in the Au–Pd encapsulated experiments. The presence of water in the melt was demonstrated by Fourier transform infrared (FTIR) spectroscopic analysis of one Pt–Gr experiment, indicating ~0·5 wt % H 2 O in the bulk composition. Further confirmation that H 2 O plays such a role in the Pt–Gr experiments was provided by an additional experiment performed in a Au–Pd capsule with ~10 wt % H 2 O specifically added. In this experiment immiscible carbonate and silicate melts were observed. Carbonate–silicate liquid immiscibility is considered to occur as a result of the H 2 O present in the system. These results can be applied to natural systems in several ways. First, the presence of a small amount of either silicate melt or H 2 O-fluid in the system will act as a ‘flux’, depressing the carbonate solidus to much lower temperatures than in anhydrous systems. Second, the full trend in melt evolution from silicate-rich to carbonate-rich melts, which is also observed in inclusions in diamonds, can be explained by melting of K- and CO 2 -bearing, water-undersaturated MORB compositions. In cratonic environments low-degree silicate and immiscible silicate and carbonate melts will metasomatize the overlying mantle in different ways, producing, in the first instance, Si enrichment and crystallization of additional orthopyroxene, phlogopite, pyrope-rich garnet and consuming olivine, and, in the second case, carbonate metasomatism, with additional magnesite–dolomite, clinopyroxene and apatite. Both metasomatic styles have been described in natural peridotite xenoliths from the cratonic lithosphere.

Description: The no-arbitrage pricing of guaranteed minimum withdrawal benefits contracts results in a singular stochastic control problem that can be formulated as a Hamilton–Jacobi–Bellman (HJB) variational inequality (VI). Recently, a penalty method has been suggested for solution of this HJB VI (Dai et al. (2008), Math. Finance , 18 , 595–611). This method is very simple to implement. In this article we present a rigorous proof of the convergence of the penalty method to the viscosity solution of the HJB VI. Numerical tests of the penalty method are presented that show the experimental rates of convergence, and a discussion of the choice of the penalty parameter is also included.

Description: Sobolev-type error analysis has recently been intensively studied for radial basis function interpolation. Although the results have been very successful, some limitations have been found. First, the spaces of target functions are not large enough for thecase $$1\le p\le \mathrm{\infty }$$ to be used practically in some applications. Second, error estimates are confined to the case of finitely smooth radial basis functions. Thus, the primary goal of this paper is to provide Sobolev-type $${L}_{p}$$ -error bounds ( $$1\le p\le \mathrm{\infty }$$ ) to functions in fractional Sobolev spaces for a wide class of radial functions including some infinitely smooth radial functions. Here an infinitely smooth radial function is required to be conditionally positive definite of a certain order $$m 〉 0$$ . In addition we provide numerical results that illustrate our theoretical error bounds.

Description: We provide new insights into the geochemistry of serpentinites from mid-ocean ridges (Mid-Atlantic Ridge and Hess Deep), passive margins (Iberia Abyssal Plain and Newfoundland) and fore-arcs (Mariana and Guatemala) based on bulk-rock and in situ mineral major and trace element compositional data collected on drill cores from the Deep Sea Drilling Project and Ocean Drilling Program. These data are important for constraining the serpentinite-hosted trace element inventory of subduction zones. Bulk serpentinites show up to several orders of magnitude enrichments in Cl, B, Sr, U, Sb, Pb, Rb, Cs and Li relative to elements of similar compatibility during mantle melting, which correspond to the highest primitive mantle-normalized B/Nb, B/Th, U/Th, Sb/Ce, Sr/Nd and Li/Y among subducted lithologies of the oceanic lithosphere (serpentinites, sediments and altered igneous oceanic crust). Among the elements showing relative enrichment, Cl and B are by far the most abundant with bulk concentrations mostly above 1000 µg g – 1 and 30 µg g – 1 , respectively. All other trace elements showing relative enrichments are generally present in low concentrations (µg g – 1 level), except Sr in carbonate-bearing serpentinites (thousands of µg g – 1 ). In situ data indicate that concentrations of Cl, B, Sr, U, Sb, Rb and Cs are, and that of Li can be, increased by serpentinization. These elements are largely hosted in serpentine (lizardite and chrysotile, but not antigorite). Aragonite precipitation leads to significant enrichments in Sr, U and B, whereas calcite is important only as an Sr host. Commonly observed brucite is trace element-poor. The overall enrichment patterns are comparable among serpentinites from mid-ocean ridges, passive margins and fore-arcs, whereas the extents of enrichments are often specific to the geodynamic setting. Variability in relative trace element enrichments within a specific setting (and locality) can be several orders of magnitude. Mid-ocean ridge serpentinites often show pronounced bulk-rock U enrichment in addition to ubiquitous Cl, B and Sr enrichment. They also exhibit positive Eu anomalies on chondrite-normalized rare earth element plots. Passive margin serpentinites tend to have higher overall incompatible trace element contents than mid-ocean ridge and fore-arc serpentinites and show the highest B enrichment among all the studied serpentinites. Fore-arc serpentinites are characterized by low overall trace element contents and show the lowest Cl, but the highest Rb, Cs and Sr enrichments. Based on our data, subducted dehydrating serpentinites are likely to release fluids with high B/Nb, B/Th, U/Th, Sb/Ce and Sr/Nd, rendering them one of the potential sources of some of the characteristic trace element fingerprints of arc magmas (e.g. high B/Nb, high Sr/Nd, high Sb/Ce). However, although serpentinites are a substantial part of global subduction zone chemical cycling, owing to their low overall trace element contents (except for B and Cl) their geochemical imprint on arc magma sources (apart from addition of H 2 O, B and Cl) can be masked considerably by the trace element signal from subducted crustal components.

Description: Carbonates carried by oceanic plates—in the form of sediment or alteration products of basalts—are introduced into the mantle by subduction. The high-pressure–high-temperature stability of carbonates in a closed system has been constrained by a number of studies, but the effects of the interactions between subducted carbonates and the surrounding mantle on carbonate stability are poorly known. These interactions may, however, influence the stability depth of the subducted material, the composition of the interaction zone, and the rate of carbon transfer from the slab to the mantle. To determine the exchange mechanisms at the interface between a subducted dolomite + coesite assemblage and the overlying mantle, we performed experiments at 6 GPa and 900°C and 1100°C in a multi-anvil apparatus. In a first series of experiments, we studied the reaction and migration processes operating along the interface between a cylinder of dolomite + coesite and a cylinder of garnet peridotite. In a second series of experiments, homogeneous mixtures of dolomite + coesite and garnet peridotite were equilibrated at high pressure and high temperature to characterize the phase relations as a function of the (dolomite + coesite)/peridotite ratio. We show that the destabilization temperature of a subducted dolomite + coesite assemblage is less than 900°C at 6 GPa when in contact with a garnet lherzolite or a harzburgite. A reaction zone composed mainly of clinopyroxene + magnesite is produced at the interface between dolomite + coesite and peridotite. Carbon—in the form of a fluid or in a carbonatitic melt—also infiltrates the garnet peridotite to form magnesite and clinopyroxene. Moreover, graphite was observed in two experiments. We believe that it was produced by a redox reaction at the interface between dolomite + coesite and iron-bearing silicates in the peridotite. As our experimental conditions are in the stability field of diamond, this suggests a potential mechanism for diamond crystallization from subducted dolomite + coesite in contact with the mantle wedge.

Description: Taranaki (Mt. Egmont) in the western North Island of New Zealand is a high-K andesite volcano with an eruptive history extending over more than 200 kyr. In general, petrological research has concentrated on the post-10 ka record of the modern edifice. This study focuses on the earlier history, which is recorded in 11 major pre-7 ka debris avalanche deposits. Each of these formed as a result of a catastrophic collapse of the edifice of the time. The clast assemblages of these deposits provide insights into the chemical compositions of magmas erupted during the earlier stages of activity of the volcano and form the basis for a new chemo-stratigraphic analysis of the pre-10 ka volcanic succession. Sample suites from the studied debris avalanche deposits show a progressive enrichment in K 2 O and large ion lithophile elements (LILE), reflecting a gradual evolution to high-K andesite. The early magmatic system (pre-100 ka) produced a wide range of compositions including relatively primitive basalts and basaltic andesites. These rocks contain phenocryst assemblages that indicate crystallization within the lower crust or mantle, including a broad range of clinopyroxene compositions, high-Al 2 O 3 hornblende, olivine and phlogopite. A higher proportion of high-silica compositions in the younger sample suites and the appearance of late-stage, low-pressure mineral phases, such as high-TiO 2 hornblende, biotite and Fe-rich orthopyroxene, reflect a gradual shift to more evolved magmas with time. These new data are interpreted to reflect a multi-stage origin for Taranaki andesites. Parental magmas were generated within a lower crustal ‘hot zone’, which formed as a result of repeated intrusions of primitive melts into the lower crust. The geochemical and mineralogical evidence indicates that prior to 100 ka this zone was relatively thin and cold, so that primitive magmas were able to rise rapidly through the crust without significant interaction and modification. As the hot zone evolved, larger proportions of intruded and underplated mafic material were partially remelted, and interaction of these melts with fractionating mantle-derived magmas generated progressively more K- and LILE-enriched compositions. A complex and dispersed magma assembly and storage system developed in the upper crust where the hot-zone melts were further modified by fractional crystallization and magma mixing and mingling.