Partial melting in an upwelling mantle column

Decompression melting of hot upwelling rock in the mantle creates a region of partial melt comprising a porous solid matrix through which magma rises buoyantly. Magma transport and the compensating matrix deformation are commonly described by two-phase compaction models, but melt production is less often incorporated. Melting is driven by the necessity to maintain thermodynamic equilibrium between mineral grains in the partial melt; the position and amount of partial melting that occur are thus thermodynamically determined. We present a consistent model for the ascent of a one-dimensional column of rock and provide solutions that reveal where and how much partial melting occurs, the positions of the boundaries of the partial melt being determined by conserving energy across them. Thermodynamic equilibrium of the boundary between partial melt and the solid lithosphere requires a boundary condition on the effective pressure (solid pressure minus melt pressure), which suggests that large effective stresses, and hence fracture, are likely to occur near the base of the lithosphere. Matrix compaction, melt separation and temperature in the partially molten region are all dependent on the effective pressure, a fact that can lead to interesting oscillatory boundary-layer structures. © 2008 The Royal Society

Englacial Pore Water Localizes Shear in Temperate Ice Stream Margins

The margins of fast‐moving ice streams are characterized by steep velocity gradients. Some of these gradients cannot be explained by a temperature‐dependent viscosity alone. Laboratory data suggest that water in the ice‐grain matrix decreases the ice viscosity; we propose that this causes the strong localization of shear in temperate ice stream margins. However, the magnitude of weakening and its consequences for ice stream dynamics are poorly understood. Here we investigate how the coupling between temperate ice properties, ice mechanics, and drainage of melt water from the ice stream margin alters the dynamics of ice streams. We consider the steady‐state ice flow, temperature, water content, and subglacial water drainage in an ice stream cross section. Temperate ice dynamics are modeled as a two‐phase flow, with gravity‐driven water transport in the pores of a viscously compacting and deforming ice matrix. We find that the dependence of ice viscosity on meltwater content focuses the temperate ice region and steepens the velocity gradients in the ice stream margin. It provides a possible explanation for the steep velocity gradients observed in some ice stream shear margins. This localizes heat dissipation there, which in turn increases the amount of meltwater delivered to the ice stream bed. This process is controlled by the permeability of the temperate ice and the sensitivity of ice viscosity to meltwater content, both of which are poorly constrained properties

Magmatic intrusions control Io's crustal thickness

Io, the most volcanically active body in the solar system, loses heat through eruptions of hot lava. Heat is supplied by tidal heating and is thought to be transferred through the mantle by magmatic segregation, a mode of transport that sets it apart from convecting terrestrial planets. We present a model that couples magmatic transport of tidal heat to the volcanic system in the crust, in order to determine the controls on crustal thickness, magmatic intrusions, and eruption rates. We demonstrate that magmatic intrusions are a key component of Io's crustal heat balance; around 80% of the magma delivered to the base of the crust must be emplaced and frozen as plutons to match rough estimates of crustal thickness. As magma ascends from a partially molten mantle into the crust, a decompacting boundary layer forms, which can explain inferred observations of a high-melt-fraction region.Comment: Accepted to JGR:Planets. 24 pages inc appendices and references. 7 figure

Homogenized boundary conditions and resonance effects\ud in Faraday cages

We present a mathematical study of two-dimensional electrostatic and electromagnetic shielding by a cage of conducting wires (the so-called `Faraday cage effect'). Taking the limit as the number of wires in the cage tends to infinity we use the asymptotic method of multiple scales to derive continuum models for the shielding, involving homogenized boundary conditions on an effective cage boundary. We show how the resulting models depend on key cage parameters such as the size and shape of the wires, and, in the electromagnetic case, on the frequency and polarisation of the incident field. In the electromagnetic case there are resonance effects, whereby at frequencies close to the natural frequencies of the equivalent solid shell, the presence of the cage actually amplifies the incident field, rather than shielding it. By appropriately modifying the continuum model we calculate the modified resonant frequencies, and their associated peak amplitudes. We discuss applications to radiation containment in microwave ovens and acoustic scattering by perforated shells

A dynamical systems approach to the tilted Bianchi models of solvable type

We use a dynamical systems approach to analyse the tilting spatially homogeneous Bianchi models of solvable type (e.g., types VI$_h$ and VII$_h$) with a perfect fluid and a linear barotropic $\gamma$-law equation of state. In particular, we study the late-time behaviour of tilted Bianchi models, with an emphasis on the existence of equilibrium points and their stability properties. We briefly discuss the tilting Bianchi type V models and the late-time asymptotic behaviour of irrotational Bianchi VII$_0$ models. We prove the important result that for non-inflationary Bianchi type VII$_h$ models vacuum plane-wave solutions are the only future attracting equilibrium points in the Bianchi type VII$_h$ invariant set. We then investigate the dynamics close to the plane-wave solutions in more detail, and discover some new features that arise in the dynamical behaviour of Bianchi cosmologies with the inclusion of tilt. We point out that in a tiny open set of parameter space in the type IV model (the loophole) there exists closed curves which act as attracting limit cycles. More interestingly, in the Bianchi type VII$_h$ models there is a bifurcation in which a set of equilibrium points turn into closed orbits. There is a region in which both sets of closed curves coexist, and it appears that for the type VII$_h$ models in this region the solution curves approach a compact surface which is topologically a torus.Comment: 29 page

The Asymptotic Behaviour of Tilted Bianchi type VI0_0 Universes

We study the asymptotic behaviour of the Bianchi type VI$_0$ universes with a tilted $\gamma$-law perfect fluid. The late-time attractors are found for the full 7-dimensional state space and for several interesting invariant subspaces. In particular, it is found that for the particular value of the equation of state parameter, $\gamma=6/5$, there exists a bifurcation line which signals a transition of stability between a non-tilted equilibrium point to an extremely tilted equilibrium point. The initial singular regime is also discussed and we argue that the initial behaviour is chaotic for $\gamma<2$.Comment: 22 pages, 4 figures, to appear in CQ

A Mathematical Model for Flash Sintering

A mathematical model is presented for the Joule heating that occurs in a ceramic powder compact during the process of flash sintering. The ceramic is assumed to have an electrical conductivity that increases with temperature, and this leads to the possibility of runaway heating that could facilitate and explain the rapid sintering seen in experiments. We consider reduced models that are sufficiently simple to enable concrete conclusions to be drawn about the mathematical nature of their solutions. In particular we discuss how different local and non-local reaction terms, which arise from specified experimental conditions of fixed voltage and current, lead to thermal runaway or to stable conditions. We identify incipient thermal runaway as a necessary condition for the flash event, and hence identify the conditions under which this is likely to occur.Comment: 14 pages, 9 figure

Cosmic No Hair for Collapsing Universes

It is shown that all contracting, spatially homogeneous, orthogonal Bianchi cosmologies that are sourced by an ultra-stiff fluid with an arbitrary and, in general, varying equation of state asymptote to the spatially flat and isotropic universe in the neighbourhood of the big crunch singularity. This result is employed to investigate the asymptotic dynamics of a collapsing Bianchi type IX universe sourced by a scalar field rolling down a steep, negative exponential potential. A toroidally compactified version of M*-theory that leads to such a potential is discussed and it is shown that the isotropic attractor solution for a collapsing Bianchi type IX universe is supersymmetric when interpreted in an eleven-dimensional context.Comment: Extended discussion to include Kantowski-Sachs universe. In press, Classical and Quantum Gravit

On the evolution of a large class of inhomogeneous scalar field cosmologies

The asymptotic behaviour of a family of inhomogeneous scalar field cosmologies with exponential potential is studied. By introducing new variables we can perform an almost complete analysis of the evolution of these cosmologies. Unlike the homogeneous case (Bianchi type solutions), when k^2<2 the models do not isotropize due to the presence of the inhomogeneitiesComment: 23 pages, 1 figure. Submitted to Classical and Quantum Gravit

Self-similar Bianchi models: II. Class B models

In a companion article (referred hearafter as paper I) a detailed study of the simply transitive Spatially Homogeneous (SH) models of class A concerning the existence of a simply transitive similarity group has been given. The present work (paper II) continues and completes the above study by considering the remaining set of class B models. Following the procedure of paper I we find all SH models of class B subjected only to the minimal geometric assumption to admit a proper Homothetic Vector Field (HVF). The physical implications of the obtained geometric results are studied by specialising our considerations to the case of vacuum and $\gamma -$law perfect fluid models. As a result we regain all the known exact solutions regarding vacuum and non-tilted perfect fluid models. In the case of tilted fluids we find the \emph{general }self-similar solution for the exceptional type VI$_{-1/9}$ model and we identify it as equilibrium point in the corresponding dynamical state space. It is found that this \emph{new} exact solution belongs to the subclass of models $n_\alpha ^\alpha =0$, is defined for $\gamma \in (\frac 43,\frac 32)$ and although has a five dimensional stable manifold there exist always two unstable modes in the restricted state space. Furthermore the analysis of the remaining types, guarantees that tilted perfect fluid models of types III, IV, V and VII$_h$ cannot admit a proper HVF strongly suggesting that these models either may not be asymptotically self-similar (type V) or may be extreme tilted at late times. Finally for each Bianchi type, we give the extreme tilted equilibrium points of their state space.Comment: Latex, 15 pages, no figures; to appear in Classical Quantum Gravity (uses iopart style/class files); (v2) minor corrections to match published versio