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Timetable (MIMW01)

From foundations to state-of-the-art in magma/mantle dynamics

Monday 15th February 2016 to Friday 19th February 2016

Monday 15th February 2016
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from John Toland (INI Director) INI 1
10:00 to 11:00 Dan McKenzie
Melt generation and extraction from the Mantle
In 1964 Paul Gast discovered that basalts from oceanic islands had quite different radiogenic isotope ratios from those of basalts from spreading ridges, and that these differences required their two source regions to have been separate for Ga. At the time he made this discovery almost all earth scientists believed that the continents had never moved. This belief changed completely over the following five years, to be replaced by a general acceptance that the Earth's mantle is a vigorously convecting fluid with a Rayleigh number of . Gast's discovery, which was quickly confirmed by many other measurements, then slowly became recognised as more and more unexpected. How can the source regions of ocean island basalts remain intact for long periods of time in a vigorously convecting fluid, and, if they are drawn out into thin sheets, as seems almost inevitable, how can melt be extracted preferentially from such a well stirred inhomogeneous material? Attempts to answer these questions have concerned igneous petrologists and geochemists ever since Gast's discovery, and are of central concern to this workshop. Though the resulting work has lead to considerable progress in understanding the process of melt generation, and some progress in understanding how it moves, it has not yet thrown much light on the original question. 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Paul Asimow
Mantle Melting: the achievements, promise, and limits of thermodynamics
This workshop is an opportunity to consider the state of the art in application of thermodynamics to mantle melting. This approach constructs and calibrates Gibbs free energy models for relevant phases, enabling predictions of equilibrium states including phase proportions, compositions, and some physical properties. We will consider the achievements of this approach, its unfulfilled promise, and its theoretical limits.

Under achievements, a body of work has applied the MELTS and pMELTS models to upper mantle magmatism in ocean ridge, back-arc, intraplate, and subduction environments. From 0-3 GPa pressure, models of multicomponent minerals and silicate liquid appear good enough to recover most experimental results, to provide an internally-consistent forward model from source peridotite through melting to the observed volume and composition of magmas, to constrain the potential temperature of source regions, to elucidate the role of water, and more. There have been a few forays into integrating this approach with magma dynamics at various scales.

Under promise, we expect near-future, improved versions that incorporate carbon species, extend to substantially higher pressure, and account for variable speciation in the melt to achieve higher fidelity. Calibration strategies based on a flexible, evolving, community framework will liberate the enterprise from the slow progress of models based on fixed, internally-consistent datasets. Fully coupled, large-scale dynamical simulations incorporating MELTS thermodynamics are practical and close to implementation.

Under limits, we must recall the limitations of the equilibrium assumptions underlying thermodynamics. At some time and length scales these assumptions break down. In a heterogeneous mantle containing species with widely divergent mobility, there must be important roles for kinetics. Progress in incorporating kinetics in major element models has been nearly nil and must be a major focus of our efforts.

Related Links
12:30 to 13:30 Lunch @ Wolfson Court
14:30 to 15:30 John MacLennan
Observational constraints on mantle melt transport
The review paper of Kelemen et al. (1997) summarised the key observations used to develop a model of rapid melt extraction in porous channels. Many subsequent observations have been acquired with the aim of refining estimates of the timescales of melt extraction from the mantle. Nevertheless, substantial uncertainty remains concerning how these observations can be used to characterise variation in melt transport velocities under basaltic volcanoes at spreading ridges and ocean islands. Uranium series disequilibria indicate that melt transport rates are almost certainly >1 m yr-1 and probably >50 m yr-1. An alternative method for estimating melt transport rates is based upon the observational record of the response of magmatism to loading and unloading of a mantle melting region. Such loading cycles may be generated by with glaciation and sea-level change. It is well-established that the record of volcanic productivity in certain regions of Iceland is strongly influenced by variations in the load from glaciation of the island. The minimal time-lag between deglaciation and a burst in volcanic productivity indicates that melt transport is rapid (>50 m yr-1). More recent analyses of seafloor morphology and temporal variation in hydrothermal activity at submerged mid-ocean ridges have made a tentative link between sea-level fall during glacial growth and increased magmatic activity during glacial terminations. This link can only work if melt extraction rates are
A number of models of melt extraction have been developed in order to understand the emergence of rapid channelized melt flow. One promising research direction is the study of the behaviour of mantle where fusible heterogeneities sit in a more refractory matrix.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Neil Ribe
Rothschild Lecture: The Hawaiian Plume: What do Surface Observables Tell Us?
Co-authors: Ulrich R. Christensen (MPI fuer Sonnensystemforschung, Göttingen, Germany), L. Cserepes (Eötvös University, Budapest, Hungary (deceased)), N. Asaadi (Institute for Advanced Studies in Basic Sciences, Zanjan, Iran), F. Sobouti (Institute for Advanced Studies in Basic Sciences, Zanjan, Iran)

The Hawaiian islands are the best known and most studied example of intraplate volcanoes generated by melting in an ascending mantle plume. Fluid mechanical models allow us to explore the link between the intrinsic physical properties of the plume and a variety of signatures that can be observed at the surface. We use two complementary models : a full three-dimensional (3-D) model with temperature- and pressure-dependent viscosity, and an isothermal “lubrication theory” model. The full 3-D model allows surface observables to be calculated directly, while the lubrication model reveals the dynamical scaling laws obeyed by those observables. Because evaluating the scaling laws is 10^9 times faster than running the 3-D model, we can invert them to determine the ranges of plume properties that are compatible with the observed values of the height of the Hawaiian swell, its width, and the total melting rate in the plume. Turning to the geoid anomaly, we discuss recent work showing that the geoid/topography ratio decreases steadily to the northwest, indicating that the plume thins the overriding lithosphere significantly . Finally, we study an extended lubrication model with a non-Newtonian rheology, which predicts a swell shape that agrees better with observations than a Newtonian model. 
17:00 to 18:00 Welcome Wine Reception
Tuesday 16th February 2016
09:00 to 10:00 John Rudge
An overview of the two-phase-flow equations for magma dynamics
The equations of two-phase-flow arise from statements of conservation of mass, momentum, and energy. In addition to the conservation laws, a series of phenomenological laws must be prescribed to describe the interaction between the two phases. It is the choice of these phenomenological laws that makes two-phase-flow theory challenging.

In this presentation I will outline the choices of phenomenological laws that have been used thus far in magma dynamics, and their physical consequences. I will give an overview of the basic physics of compaction, and the important role of the compaction length, the natural length-scale in compaction problems. I will highlight which areas of the theory seem robust, and which are in need of further development.
10:00 to 11:00 Marc Spiegelman
The classics: Understanding the basic behavior of the "McKenzie" Equations
Given a set of consistent conservation equations for two-phase flow in deformable porous media, this presentation strives to develop a better physical intuition into their behavior through a series of classical model problems. These problems illustrate the basic behavior of the equations as well as highlighting some of the open questions and continuing mathematical/computational issues involved with their solution.

We begin by emphasizing that the solid velocity field can be usefully decomposed into incompressible and compressible components. Incompressible flow is governed by Stokes equation for porosity driven convection, while compressible flow is governed by a non-linear wave equation for the evolution of porosity. The driving force for non-linear waves is the non-linearity between porosity and permeability. However, the waveforms and propagation speed depend on the solid bulk rheology. Depending on choice of solid rheology, these waves can manifest as kinematic shocks, non-local dispersive solitary waves and wave-trains (poro-viscous) or diffusive porosity waves (poro-elastic). However, these waves all arise to accommodate variations in melt flux on scales much larger than the compaction length. We will also consider other localization phenomena that occur on scales smaller than the compaction length including the development of melt-bands due to shear of a solid with poro sity weakening shear viscosity, and channel formation due to simplified reactive flow.

These equations demonstrate a rich array of behavior, all arising from the rheological response of a deformable solid to variations in fluid flux. These solutions suggest that magma transport in the mantle is strongly localized and time-dependent. While considerable uncertainties in the formulations remain, the general framework is computationally tractable and provides a quantitative framework for investigating coupled fluid/solid mechanics in the Earth.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Sash Hier-Majumder
Partial Melting Above the Mantle Transition Zone
Recent seismic observations indicate the presence of a low velocity layer above the mantle transition zone. Marked by a reduction of shear wave speed, the top of this layer is generally detected around a depth of 350 km. In this talk, I will present an analysis of seismic data quantifying the amount of melt present in this region, stability of this partially molten layer over geologic time, and the possible origin of the layer in the context of crystallization of the Earth's mantle from a magma ocean.
12:30 to 13:30 Lunch @ Wolfson Court
14:30 to 15:30 Ian Hewitt
Two-phase-flow dynamics in ice sheets
Co-author: Christian Schoof (University of British Columbia)

The dynamics of large ice sheets have many similarities with mantle and magma dynamics. Polycrystalline material undergoes intense shear and creep deformation, slow evolution of grain structure, and dissipative heating that can cause partial melting. The presence of melt (water) can enhance creep and facilitates sliding past bedrock and sediments, leading to extreme patterns of ice flow, most clearly demonstrated in the Antarctic and Greenland ice streams.

In this talk I will review this related application of the two-phase flow equations often used to describe melt in the mantle. I will focus especially on polythermal ice, when the quantities of interest are the heat and water content together with their transport, and on the subglacial drainage system, which is thought to exhibit channelized flow similar (perhaps) to the mantle. Approximations used in the ice-sheet literature will be discussed, along with numerical solution methods for the resulting models. I will highlight analogies and difference with mantle models.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 S. Majid Hassanizadeh
Advanced theories of two-phase flow in deformable porous media, including fluid-fluid interfaces
Two-phase flow in porous media is traditionally modeled using a modified form of Darcy’s law, two volume balance equations, and a so-called capillary pressure-saturation relationship. Darcy’s Law was proposed more than 150 years ago for the flow of a single fluid in soil. Since then, this equation, in almost original form, has been assumed to be applicable to more and more complicated porous media. But, there are many shortcomings of the so-called extended Darcy’s Law. The general understanding is that capillary pressure is equal to the difference in pressures of two fluids. It is assumed to be an algebraic empirical function of saturation. This empirical relationship for capillarity has been studied extensively in soil physics, subsurface hydrology, and petroleum engineering, because of its central role in multiphase-flow theory. Yet, the standard theory of capillarity has a mostly empirical character. The macroscale capillary pressure-saturation relationsh ip cannot be derived from basic physical principles or using averaging methods. Moreover, it is known to be hysteretic (i.e., it depends on the history of the fluids’ distribution) and rate-dependent (i.e., it depends on the rate of flow or rate of change of saturation). We present a new theory of two-phase flow, which comprises a truly extended Darcy’s law and a new capillarity theory, which has four main features: i) pressure gradient is not the only driving force, ii) capillary pressure is an intrinsic property of the porous medium and is not only a function of saturation but also fluid-fluid specific interfacial areas, iii) similarly, effective stress parameter is a function of fluid-fluid specific interfacial area as well as saturation, and iv) there is a dynamic (or non-equilibrium) capillarity effect. We provide experimental evidences for the validity of the new theory. Also, pore-scale and continuum-scale simulations are used to study the possible significance of the new theory at various scales.

Related Links
Wednesday 17th February 2016
09:00 to 10:00 David Bercovici
Interface effects in two-phase phase physics
Most theories of two-phase compaction theory converge to the same set of governing equations for geologically relevant limits.  However not many include the infuence of surface energyand thermodynamics on the interface between phases.  The effect of interface surface energyhas multiple effects in two-phase flows including capillary forces and wetting, self-separation, and is even at the heart of void-generating damage theories.  Moreover, surface energy effectsdrive coarsening in mixtures and grained media (e.g., Ostwald ripening or normal grain-growth), nucleation during phase changes, and blocking or Zener pinning of migrating grain boundaries.  Surface thermodynamics is, again, central to grain-damage theories, and the combined influence ofgrain-growth, interface coarsening, Zener pinning and damage in two phases allows a reasonable model for lithospheric localization and plate tectonic generation. Finally, a new theory for diffuse interface capillary effects might provide a means for band width selection in melt-shear band instabilities.
10:00 to 11:00 Viktoriya Yarushina
(De)compaction waves in porous viscoelastoplastic media and focused fluid flow
Co-authors: Yuri Podladchikov (Institut des Sciences de la Terre, University of Lausanne), Ludovic Raess (Institut des Sciences de la Terre, University of Lausanne), Nina Simon (Institute for Energy Technology)

Understanding of tectonic processes and melt transport in the deep Earth and management of commercially valuable hydrocarbon resources and engineering operations such as CCS in the shallow Earth are dependent on our knowledge about interaction between rocks and fluids. Rock-fluid interaction involves heat and mass transfer, deformation, hydrodynamic flow and chemical reactions, thereby necessitating its consideration as a complex process coupling several simultaneous mechanisms. Two most widely used models for coupled fluid flow and rock deformation are poroelastic theory of Biot and poroviscous model of McKenzie. Even though many extensions of these two theories were proposed in recent years, there is still a considerable effort directed towards formulation of new two-phase theories. Here, we present a simple mathematical model for nonlinear porous viscoelastoplastic materials. Our main motivation is to establish a simple unifying theory for porous fluid flow in a deformable matrix that is able to capture the range of rheological responses expected within the Earth. These responses can vary from elastic small strain consolidation to plastic porosity collapse from tenths to a few percents of porosity in near-surface sediments and up to high-temperature creep during extraction of melts and metamorphic fluids. Our model is thermodynamically consistent and can be generalized to the more complex multi-phase multi-component systems. Developed framework we apply to study focused fluid flow systems that are often evidenced as dikes, veins or volcanic diatremes in the deep Earth and fluid escape pipes, gas conducting chimneys, mud volcanoes, sand injectites or pockmarks in the shallow Earth. The particular attention is to porosity waves as a flow focusing mechanism.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Peter Driscoll
Influence of mantle melting on cooling and volatile cycling
Heat lost by melt eruption at the surface is a minor component of Earth’s internal heat budget and normally discarded from thermal history models. However, the strong temperature dependence of global melt production implies that an earlier, hotter Earth produced more melt, increasing internal heat loss via eruptive cooling. To address this we develop simple models for mantle melt production and eruptive cooling, and derive melt heat loss scaling laws. Including a melt heat sink in thermal history models results in more efficient mantle cooling, exacerbating the thermal catastrophe of the mantle, which can be overcome by adjustments to the present-day energy balance. In application to stagnant lid planets, cooling by melt eruption can compensate for slow conductive cooling but is dependent on the fraction of extrusive to intrusive melt. In the same vein, a generalized volatile cycling box-model illustrates how volatile reservoir size depends on melt eruption rates.
12:30 to 13:30 Lunch @ Wolfson Court
19:30 to 22:00 Formal dinner at Emmanuel College
Thursday 18th February 2016
09:00 to 10:00 Anne Davaille
Thermal convection in an heterogeneous mantle: plumes, piles, domes, and LLSVPs.
Planetary long-term cooling, as well as surface phenomena such as plate tectonics, volcanoes and earthquakes, are mainly controlled by the existence and patterns of convective motions in the solid-state mantle. The morphology and characteristics of convective patterns strongly depends on the mantle physical properties and the existence of density heterogeneities.

I shall review how the interplay of the latter with thermal convection produces mantle instabilities of widely varying morphology and time-dependence, according to the magnitude of the density anomalies and the type of mantle rheology. They could therefore explain the diversity of hot spot volcanism observed on Earth, as well as the mantle structures revealed by seismic tomography. 
10:00 to 11:00 Andrea Tommasi
Feedbacks between deformation and melts in the upper mantle
Co-authors: Katherine Higgie (CNRS & Universite Montpellier, France), Veronique Le Roux (WHOI, USA), Vincent Soustelle (University Wuhan, China), Virginie Baptiste (CNRS & Universite Montpellier, France), Alain Vauchez (CNRS & Universite Montpellier, France), Jean Louis Bodinier (CNRS & Universite Montpellier, France)

Experiments and theoretical considerations predict feedbacks between deformation and melts, namely strain localization controlled by the presence of melts and melt focusing controlled by deformation. Yet, peridotite massifs and mantle xenoliths, which record deformation and melt transport through the shallow mantle (lithosphere and lithosphere-asthenosphere boundary), feature a wide variety of relations between melts and deformation. These relations range from strong feedbacks between deformation and melt distribution to absence of any relation between the two processes. We will present an overview of the different types of relations between deformation and melts observed in the field (km to cm scale) and in thin sections (mm scale) and highlight the questions these observations bring.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Christopher MacMinn
Large deformations in soft porous materials: Squishing, swelling, and yielding
In an elastic solid, the state of stress depends on the displacement of material points from a relaxed reference state. In a poroelastic solid, the mechanics of the solid skeleton are additionally coupled to flow through the pore structure. The classical theory of linear poroelasticity captures this coupling by combining Darcy’s law with Terzaghi’s effective stress and linear elasticity in a linearized kinematic framework. This is a good model for very small deformations, but it becomes increasingly inappropriate for moderate to large deformations, which are common in the context of phenomena such as swelling and damage, and for soft materials such as gels and tissues. Here, we review the well-known theory of large-deformation poroelasticity and then consider the implications of large deformations in the context of two model problems: (1) Classical uniaxial consolidation and (2) the swelling and drying of a spherical gel.

Related Links
12:30 to 13:30 Lunch @ Wolfson Court
14:30 to 15:30 Christian Huber
Multiphase transport, flirting with the limits of continuum models
Co-authors: Salah Faroughi (Georgia Tech), Hamid Karani (Georgia Tech), Andrea Parmigiani (ETH)

Multiphase flows can be found in a wide variety of Earth and Planetary Systems. Solving for mass, momentum and energy conservation becomes difficult because they are regulated by dynamic interfaces between the different components. Here, we will discuss the merits of continuum (spatially averaged) and discrete scale (granular where interfaces are explicitly treated) approaches in the light of different applications ranging from Planetary Sciences (water on Mars), to volcanology (bubble dynamics and rheology) and finally hydrology (heat transfer in porous media). The goal of the presentation is to discuss the necessity to develop a framework consistent with the granular scale (downscaling) as well as different elements of approaches for upscaling (effective medium theory, fractional derivatives).
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Joe Dufek
Multiphase Flow in Crustal Magmatic Processes
Multiphase dynamics in crustal magmatic processes have analogy with many processes occurring in the mantle and span phenomena from slow, dense granular flows to rapidly shearing granular flows during volcanic eruptions. In this talk I will discuss these two end-members in the context of magma chamber dynamics and eruptive dynamics. In the magma dynamics case, the efficiency of the relative motion between melt and crystals produces distinct compositional trends that can be compared with the chemistry of melts and phase equilibria. This work will consider differentiation scenarios in both simplified magma reservoir geometries and those that are emergent with the successive input from intrusions in an open system. Evidence of open systems and assembly of magmatic systems incrementally are present in a range of plutonic and volcanic rocks. To evaluate the timescales of silicic magma production, a multiphase dynamics model will be discussed that includes heat transfer, phase equil ibria and relative motion between a melt and several crystal phases. A particular focus of this presentation is a comparison of dynamic processes to proxies used as chronometers. The other end-member considered are flows produced during explosive eruptions. During explosive eruptions the interstitial fluid is often a gas and the granular flow has significant inertia. I will compare and contrast granular stress concentration, entrainment, and fluid expulsion in this regime as well as discuss the prospects for integrating a range of laboratory experimental and numerical approaches.
Friday 19th February 2016
09:00 to 10:00 Marc Hesse
Pore-scale controls on core formation in planetesimals
Co-authors: Soheil Ghanbarzadeh (University of Texas at Austin), Masa Prodanovic (University of Texas at Austin)

Pore-scale melt distribution is thought to evolve towards textural equilibrium. The topology of these pore-scale melt networks is controlled by the porosity and the dihedral angle at the contact line between two solid grains and the melt. I will present three-dimensional computations of texturally equilibrated melt networks in realistic poly-crystalline materials that have been obtained from x-ray diffraction contrast tomography. Our simulations show strong hysteresis in the topology of melt networks with large dihedral angles. A percolation threshold at dihedral angles above 60 degrees is generally thought to prevent rapid core formation in planetesimals by segregation of metallic melts. However, primitive achondrites show that the incipient melt fractions are between 25% and 35% and that metallic melt is connected despite dihedral angles of approximately 90 degrees. Our simulations show that hysteresis allows such a high porosity and high dihedral angle melt network to rem ain connected during drainage of the metallic melt. This provides a mechanism for rapid core formation in planetesimals by porous flow. Only a very small melt fraction, approximately 1%, is trapped and left behind in the silicate mantle after melt segregation. This may provide an explanation for the "excess siderophile problem" in the Earth.
10:00 to 11:00 Richard Katz
Treatment of energy and composition in models of magma/mantle dynamics and implications for reactive flow
Co-authors: Tobias Keller (University of Oxford), Andrew Turner (University of Oxford), Samuel Weatherley (Geological Survey of Denmark and Greenland)

There are relatively few theoretical studies of magma/mantle dynamics that incorporate conservation of energy and species mass in a consistent way. Doing so remains a challenge, both technically in terms of solving the equations, and scientifically in terms of formulating them in a way that is both physically reasonable and sufficiently simple. In this talk I review previous and current approaches to this problem. I highlight the importance of this treatment by discussing the question of reactive channelisation of magma. Reactive channelisation is thought to be an important process in melt transport, helping to explain some observed geochemical signals. It turns out that the behaviour of reactive-flow models depends in important ways on the way that energy and composition are treated. I discuss how mantle heterogeneity and the presence of volatile elements may be essential for channelised flow.

Related Links
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Dave May
Geodynamics and Two-Phase Flow: A Computational Perspective
The system of partial differential equations used in geodynamic contexts to study melt migration define the evolution of a two-phase system comprised of a highly viscous, porous crystalline mantle rock ("solid" phase) through which low viscosity magma ("fluid" phase) can flow. Typical geodynamic two-phase flow scenarios are characterized by rheological non-linearities, localization and flows possessing a range of characteristic length scales.

From a computational stand point, the inherently stiff and non-linear coupling between the fluid and solid phases, together with the typical flow characteristics associated with geodynamic processes represent a number of challenges with respect to: (i) the choice of primary variables; (ii) the choice of temporal and spatial discretisation scheme, and (iii) the development of efficient techniques for solving the discrete system of equations.

Within the context of these three themes, I will review the commonly adopted philosophies, computational methods and software used by the geodynamics community to study the dynamics two-phase flow systems. Trade-offs and compromises associated with the scope of the modelling software will be discussed, along with an identification of the numerical components which are presently lacking, or devoid of, rigorous mathematical analysis. Lastly, I will discuss recent algorithmic and software developments of highly efficient and scalable solver-discretisation components which will be crucial in enabling the simulation of high resolution, three-dimensional two-phase flow systems. 
12:30 to 13:30 Lunch @ Wolfson Court
14:30 to 15:30 Matthew Knepley
Computational Considerations for Magma Dynamics Simulation
Co-author: Tobin Issac, University of Chicago

The optimal solver for a given problem depends not only on the equations being solved, but the boundary conditions, discretization, parameters, problem regime, and machine architecture. This interdependence means that \textit{a priori} selection of a solver is a fraught activity and should be avoided at all costs. While there are many packages which allow flexible selection and (some) combination of linear solvers, this understanding has not yet penetrated the world of nonlinear solvers. We will briefly discuss techniques for combining nonlinear solvers, theoretical underpinnings, and show concrete examples from magma dynamics.    

The same considerations which are present for solver selection should also be taken into account when choosing a discretization. However, scientific software seems even less likely to allow the user freedom here than in the nonlinear solver regime. We will discuss tradeoffs involved in choosing a discretization of the magma dynamics problem, and demonstrate how a flexible mechanism might work using examples from the PETSc libraries from Argonne National Laboratory.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Closing Remarks by Organisers INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons