Analytical solutions of compacting flow past a sphere

AbstractA series of analytical solutions are presented for viscous compacting flow past a rigid impermeable sphere. The sphere is surrounded by a two-phase medium consisting of a viscously deformable solid matrix skeleton through which a low-viscosity liquid melt can percolate. The flow of the two-phase medium is described by McKenzie’s compaction equations, which combine Darcy flow of the liquid melt with Stokes flow of the solid matrix. The analytical solutions are found using an extension of the Papkovich–Neuber technique for Stokes flow. Solutions are presented for the three components of linear flow past a sphere: translation, rotation and straining flow. Faxén laws for the force, torque and stresslet on a rigid sphere in an arbitrary compacting flow are derived. The analytical solutions provide instantaneous solutions to the compaction equations in a uniform medium, but can also be used to numerically calculate an approximate evolution of the porosity over time whilst the porosity variations remain small. These solutions will be useful for interpreting the results of deformation experiments on partially molten rocks.This work was support by NERC standard grant NE/I023929/1.This is the author accepted manuscript. The final version is available from Cambridge University Press via https://doi.org/10.1017/jfm.2014.10

Melt-band instabilities with two-phase damage

Deformation experiments on partially molten rocks in simple shear form melt bands at 20◦ to the shear plane instead of at the expected 45◦ principal compressive stress direction. Thesemelt bands may play an important role in melt focusing in mid-ocean ridges. Such shallow bands are known to form for two-phase media under shear if strongly non-Newtonian power-law creep is employed for the solid phase, or anisotropy imposed. However laboratory experiments show that shallow bands occur regardless of creep mechanism, even in diffusion creep, which is nominally Newtonian. Here we propose that a couple of forms of two-phase damage allow for shallow melt bands even in diffusion creep.Support was provided by the National Science Foundation (NSF, grant EAR-1015229), the Natural Environment Research Council (NERC, grant NE/I023929/1) and Trinity College.This is the final published version. This article has been accepted for publication in Geophysical Journal International ©: 2015 the Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved

A mechanism for mode selection in melt band instabilities

The deformation of partially molten mantle in tectonic environments can lead to exotic structures, which potentially affect both melt and plate-boundary focussing. Examples of such structures are found in laboratory deformation experiments on partially molten rocks. Simple-shear and torsion experiments demonstrate the formation of concentrated melt bands at angles of around 20° to the shear plane. The melt bands form in the experiments with widths between a few to tens of microns, and a band spacing roughly an order of magnitude larger. Existing compaction theories, however, cannot predict this band width structure, let alone any mode selection, since they infer the fastest growing instability to occur for wavelengths or bands of vanishing width. Here, we propose that surface tension in the mixture, especially on a diffuse interface in the limit of sharp melt-fraction gradients, can mitigate the instability at vanishing wavelength and thus permit mode selection for finite-width bands. Indeed, the expected weak capillary forces on the diffuse interface lead to predicted mode selection at the melt-band widths observed in the experiments.The authors sincerely thank Sam Butler and Ben Holtzman for thoughtful reviews. Support was provided by the National Science Foundation (NSF, grant EAR-1344538), the Natural Environment Research Council (NERC, grant NE/I023929/1) and Trinity College.This is the final version of the article. It first appeared from Elsevier via http://dx.doi.org/10.1016/j.epsl.2015.10.05

Fast magma ascent, revised estimates from the deglaciation of Iceland

Partial melting of asthenospheric mantle generates magma that supplies volcanic systems. The timescale of melt extraction from the mantle has been hotly debated. Microstructural measurements of permeability typically suggest relatively slow melt extraction (1 m/yr) whereas geochemical (Uranium-decay series) and geophysical observations suggest much faster melt extraction (100 m/yr). The deglaciation of Iceland triggered additional mantle melting and magma flux at the surface. The rapid response has been used to argue for relatively rapid melt extraction. However, this episode must, at least to some extent, be unrepresentative, because the rates of magma eruption at the surface increased about thirty-fold relative to the steady state. Our goal is to quantify this unrepresentativeness. We develop a one-dimensional, time-dependent and nonlinear (far from steady-state), model forced by the most recent, and best mapped, Icelandic deglaciation. We find that 30 m/yr is the best estimate of the steady-state maximum melt velocity. This is a factor of about 3 smaller than previously claimed, but still relatively fast. We translate these estimates to other mid-ocean ridges accounting for differences in passive and active upwelling and degree of melting. We find that fast melt extraction greater than about 10 m/yr prevails globally.Leverhulme Trus

Gravity, Topography, and Melt Generation Rates From Simple 3-D Models of Mantle Convection

Convection in fluid layers at high Rayleigh number (Ra $\sim 10^6$) have a spoke pattern planform. Instabilities in the bottom thermal boundary layer develop into hot rising sheets of fluid, with a component of radial flow towards a central upwelling plume. The sheets form the "spokes" of the pattern, and the plumes the "hubs". Such a pattern of flow is expected to occur beneath plate interiors on Earth, but it remains a challenge to use observations to place constraints on the convective planform of the mantle. Here we present predictions of key surface observables (gravity, topography, and rates of melt generation) from simple 3D numerical models of convection in a fluid layer. These models demonstrate that gravity and topography have only limited sensitivity to the spokes, and mostly reflect the hubs (the rising and sinking plumes). By contrast, patterns of melt generation are more sensitive to short wavelength features in the flow. There is the potential to have melt generation along the spokes, but at a rate which is relatively small compared with that at the hubs. Such melting of spokes can only occur when the lithosphere is sufficiently thin ($\lesssim 80$ km) and mantle water contents are sufficiently high ($\gtrsim 100$ ppm). The distribution of volcanism across the Middle East, Arabia and Africa north of equator suggests that it results from such spoke pattern convection.Leverhulme Trus

The feasibility of thermal and compositional convection in Earth's inner core

Inner core convection, and the corresponding variations in grain size and alignment, has been proposed to explain the complex seismic structure of the inner core, including its anisotropy, lateral variations and the F-layer at the base of the outer core. We develop a parametrized convection model to investigate the possibility of convection in the inner core, focusing on the dominance of the plume mode of convection versus the translation mode. We investigate thermal and compositional convection separately so as to study the end-members of the system. In the thermal case the dominant mode of convection is strongly dependent on the viscosity of the inner core, the magnitude of which is poorly constrained. Furthermore recent estimates of a large core thermal conductivity result in stable thermal stratification, hindering convection. However, an unstable density stratification may arise due to the pressure dependant partition coefficient of certain light elements. We show that this unstable stratification leads to compositionally driven convection, and that inner core translation is likely to be the dominant convective mode due to the low compositional diffusivity. The style of convection resulting from a combination of both thermal and compositional effects is not easy to understand. For reasonable parameter estimates, the stabilizing thermal buoyancy is greater than the destabilizing compositional buoyancy. However we anticipate complex double diffusive processes to occur given the very different thermal and compositional diffusivities.We would like to thank Chris Davies for help with comparison to his results, plus Deputy Editor Stephane Labrosse, Renaud Deguen and ´ an anonymous reviewer for constructive comments that improved the manuscript. KHL and AD are funded by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement number 204995. JAN is partially funded by a Royal Society University Research Fellowship.This is the final published version. This article has been accepted for publication in Geophysical Journal International ©: 2015 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved

Uplift histories of Africa and Australia from linear inverse modeling of drainage inventories

We describe and apply a linear inverse model which calculates spatial and temporal patterns of uplift rate by minimizing the misfit between inventories of observed and predicted longitudinal river profiles. Our approach builds upon a more general, non-linear, optimization model, which suggests that shapes of river profiles are dominantly controlled by upstream advec- tion of kinematic waves of incision produced by spatial and temporal changes in regional uplift rate. Here, we use the method of characteristics to solve a version of this problem. A damped, non-negative, least squares approach is developed that permits river profiles to be inverted as a function of up- lift rate. An important benefit of a linearized treatment is low computational cost. We have tested our algorithm by inverting 957 river profiles from both Africa and Australia. For each continent, the drainage network was constructed from a digital elevation model. The fidelity of river profiles extracted from this network was carefully checked using satellite imagery. River profiles were inverted many times to systematically investigate the trade-off between model misfit and smoothness. Spatial and temporal patterns of both uplift rate and cumulative uplift were calibrated using independent geologic and geophys- ical observations. Uplift patterns suggest that the topography of Africa and Australia grew in Cenozoic times. Inverse modeling of large inventories of river profiles demonstrates that drainage networks contain coherent signals that record the regional growth of elevation.This is the final version. It first appeared at http://onlinelibrary.wiley.com/wol1/doi/10.1002/2014JF003297/abstract

The global melt inclusion C/Ba array: Mantle variability, melting process, or degassing?

The Earth’s mantle holds more carbon than its oceans, atmosphere and con- tinents combined, yet the distribution of carbon within the mantle remains uncertain. Our best constraints on the distribution of carbon within the up- per mantle are derived from the carbon-trace element systematics of ultra- depleted glasses and melt inclusions from mid-ocean ridge basalts. How- ever, carbon-trace element systematics are susceptible to modification by crustal processes, including concurrent degassing and mixing, and melt in- clusion decrepitation. In this study we explore how the influence of these processes varies systematically with both the mantle source and melting pro- cess, thereby modulating both global and local carbon-trace element trends. We supplement the existing melt inclusion data from Iceland with four new datasets, significantly enhancing the spatial and geochemical coverage of melt inclusion datasets from the island. Within the combined Iceland dataset there is significant variation in melt inclusion C/Ba ratio, which is tightly correlated with trace element enrichment. The trends in C/Ba- Ba space displayed by our new data coincide with the same trends in data compiled from global ocean islands and mid-ocean ridges, forming a global array. The overall structure of the global C/Ba-Ba array is not a property of the source, instead it is controlled by CO2 vapour loss pre- and post-melt inclusion entrapment; i.e., the array is a consequence of degassing creating near-constant maximum melt-inclusion carbon contents over many orders of magnitude of Ba concentration. On Iceland, extremely high C/Ba (>100) and C/Nb (>1000) ratios are found in melt inclusions from the most depleted eruptions. The high C/Ba and C/Nb ratios are unlikely to be either analytical artefacts, or to be the product of extreme fractionation of the most incompatible elements during silicate melting. Whilst high C/Ba and C/Nb ratios could be generated by regassing of melt inclusions by CO2 vapour, or by mantle melting occurring in the presence of residual graphite, we suggest the high values most likely derive from an intrinsically high C/Ba and C/Nb mantle component that makes up a small fraction of the Icelandic mantle

Experimental Study of Dislocation Damping Using a Rock Analogue

In order to explore the effects of dislocations on seismic velocity and attenuation, we conducted a series of forced oscillation and ultrasonic tests on rock analogue samples (polycrystalline borneol) that were pre-deformed under various differential stress ∆σ. Additionally, creep experiments were conducted to determine the steady-state flow law for borneol. The dominant deformation mechanism of polycrystalline borneol changes from diffusion to dislocation creep at about ∆σ = 2 MPa. At high stresses, power law creep with a stress exponent of ∼4 was measured. Microstructure of the deformed samples showed wavy grain boundaries due to dislocation-induced migration, and the occasional existence of microcracks. A borneol sample deformed in the dislocation creep regime showed a significant reduction in Young’s modulus E and a slight increase in attenuation Q^−1 at frequencies lower than 100 Hz, whereas E at ultrasonic frequency (10^6 Hz) did not reduce. Therefore, a major part of the dislocation creep-induced anelastic relaxation is a peak with a characteristic frequency between 100 and 10^6 Hz, which is much higher than the range of grain boundary-induced anelasticity of this material. Further experiments under higher confining pressure are needed to assess the relative contribution from dislocations and microcracks to this peak.JSPS KAKENHI Grant Number JP15K1356

On mass transport in porosity waves

Porosity waves arise naturally from the equations describing fluid migration in ductile rocks. Here, we show that higher-dimensional porosity waves can transport mass and therefore preserve geochemical signatures, at least partially. Fluid focusing into these high porosity waves leads to recirculation in their center. This recirculating fluid is separated from the background flow field by a circular dividing streamline and transported with the phase velocity of the porosity wave. Unlike models for onedimensional chromatography in geological porous media, tracer transport in higher-dimensional porosity waves does not produce chromatographic separations between relatively incompatible elements due to the circular flow pattern. This may allow melt that originated from the partial melting of fertile heterogeneities or fluid produced during metamorphism to retain distinct geochemical signatures as they rise buoyantly towards the surface