Richard F. Katz

The dynamics of melt and shear localization in partially molten aggregates

The volcanoes that lie along the Earths tectonic boundaries are fed by melt generated in the mantle. How this melt is extracted and focused to the volcanoes, however, remains an unresolved question. Here we present new theoretical results with implications for melt focusing beneath mid-ocean ridges. By modelling laboratory experiments, we test a formulation for magma dynamics and provide an explanation for localized bands of high-porosity and concentrated shear deformation observed in experiments. These bands emerge and persist at 15°–25° to the plane of shear. Past theoretical work on this system predicted the emergence of melt bands but at an angle inconsistent with experiments. Our results suggest that the observed band angle results from a balance of porosity-weakening and strain-rate-weakening deformation mechanisms. Lower band angles are predicted for greater strain-rate weakening. From these lower band angles, we estimate the orientation of melt bands beneath mid-ocean ridges and show that they may enhance magma focusing toward the ridge axis.

Melting above the anhydrous solidus controls the location of volcanic arcs

Segregation of magma from the mantle in subduction zones is one of the principal mechanisms for chemical differentiation of the Earth. Fundamental aspects of this system, in particular the processes by which melt forms and travels to the Earth’s surface, remain obscure. Systematics in the location of volcanic arcs, the surface expression of this melting, are widely considered to be a clue to processes taking place at depth, but many mutually incompatible interpretations of this clue exist (for example, see refs 1–6). We discriminate between those interpretations by the use of a simple scaling argument derived from a realistic mathematical model of heat transfer in subduction zones. The locations of the arcs cannot be explained by the release of fluids in reactions taking place near the top of the slab. Instead, the sharpness of the volcanic fronts, together with the systematics of their locations, requires that arcs must be located above the place where the boundary defined by the anhydrous solidus makes its closest approach to the trench. We show that heat carried by magma rising from this region is sufficient to modify the thermal structure of the wedge and determine the pathway through which both wet and dry melts reach the surface.

Stability of ice-sheet grounding lines

Recent observations of the West Antarctic Ice Sheet document rapid changes in the mass balance of its component glaciers. These observations raise the question of whether changing climatic conditions have triggered a dynamical instability in the ice-sheet–ice-shelf system. The dynamics of marine ice sheets are sensitive to grounding-line position and variation, characteristics that are poorly captured by most current models. We present a theory for grounding-line dynamics in three spatial dimensions and time. Our theory is based on a balance of forces across the grounding line; it is expressed as a differential equation that is analogous to the canonical Stefan condition. We apply this theory to the question of grounding-line stability under conditions of retrograde bed slope in a suite of calculations with different basal topography. A subset of these have basal topography inspired by the Pine Island glacier, where basal depth varies in both the along-flow and across-flow directions. Our results indicate that unstable retreat of the grounding line over retrograde beds is a robust feature of models that evolve based on force balance at the grounding line. We conclude, based on our simplified model, that unstable grounding-line recession may already be occurring at the Pine Island glacier.

Porosity‐driven convection and asymmetry beneath mid‐ocean ridges

Seismic tomography of the asthenosphere beneath mid-ocean ridges has produced images of wave speed and anisotropy that are asymmetric across the ridge axis. These features have been interpreted as resulting from an asymmetric distribution of upwelling and melting. Using computational models of coupled magma/mantle dynamics beneath mid-ocean ridges, I show that such asymmetry should be expected if buoyancy forces contribute to mantle upwelling beneath ridges. The sole source of buoyancy considered here is the dynamic retention of less dense magma within the pores of the mantle matrix. Through a scaling analysis and comparison with a suite of simulations, I derive a quantitative prediction of the contribution of such buoyancy to upwelling; this prediction of convective vigor is based on parameters that, for the Earth, can be constrained through natural observations and experiments. I show how the width of the melting region and the crustal thickness, as well as the susceptibility to asymmetric upwelling, are related to convective vigor. I consider three causes of symmetry breaking: gradients in mantle potential temperature and composition and ridge migration. I also report that in numerical experiments performed for this study, the fluid dynamical instability associated with porosity/shear band formation is not observed to occur.

Glacial cycles drive variations in the production of oceanic crust

Connecting orbit to the ocean floor The amount of magma erupted at mid-ocean ridges can be modified by periodic ice ages that alter sea level. Crowley et al. analyzed high-resolution ocean depth data across the Australian-Antarctic ocean ridge (see the Perspective by Conrad). The results revealed 23-, 41-, and 100-thousand-year periodicity. These periods are similar to the well-known Milankovitch cycles associated with ice ages that are triggered by changes in Earths orbit. Decreasing sea levels decrease the overlying pressure, thereby increasing the amount of erupted magma. The cyclic nature of glaciations and sea level creates a series of spaced topographic highs along the sea floor. Thus, Earths atmosphere and mantle are coupled on a glacial time scale. Science, this issue p. 1237; see also p. 1204 Spectral analysis reveals topographic variations consistent with glacial cycles across the Australian-Antarctic ocean ridge. [Also see Perspective by Conrad] Glacial cycles redistribute water between oceans and continents, causing pressure changes in the upper mantle, with consequences for the melting of Earth’s interior. Using Plio-Pleistocene sea-level variations as a forcing function, theoretical models of mid-ocean ridge dynamics that include melt transport predict temporal variations in crustal thickness of hundreds of meters. New bathymetry from the Australian-Antarctic ridge shows statistically significant spectral energy near the Milankovitch periods of 23, 41, and 100 thousand years, which is consistent with model predictions. These results suggest that abyssal hills, one of the most common bathymetric features on Earth, record the magmatic response to changes in sea level. The models and data support a link between glacial cycles at the surface and mantle melting at depth, recorded in the bathymetric fabric of the sea floor.

Subglacial hydrology and the formation of ice streams

Antarctic ice streams are associated with pressurized subglacial meltwater but the role this water plays in the dynamics of the streams is not known. To address this, we present a model of subglacial water flow below ice sheets, and particularly below ice streams. The base-level flow is fed by subglacial melting and is presumed to take the form of a rough-bedded film, in which the ice is supported by larger clasts, but there is a millimetric water film which submerges the smaller particles. A model for the film is given by two coupled partial differential equations, representing mass conservation of water and ice closure. We assume that there is no sediment transport and solve for water film depth and effective pressure. This is coupled to a vertically integrated, higher order model for ice-sheet dynamics. If there is a sufficiently small amount of meltwater produced (e.g. if ice flux is low), the distributed film and ice sheet are stable, whereas for larger amounts of melt the ice–water system can become unstable, and ice streams form spontaneously as a consequence. We show that this can be explained in terms of a multi-valued sliding law, which arises from a simplified, one-dimensional analysis of the coupled model.

The Role of Volatiles in Reactive Melt Transport in the Asthenosphere

Experimental studies of mantle petrology find that small concentrations of water and carbon dioxide have a large effect on the solidus temperature and distribution of melting in the upper mantle. However, it has remained unclear what effect small fractions of deep, volatile-rich melts have on melt transport and reactive melting in the shallow asthenosphere. Here we present theory and computations indicating that low-degree, reactive, volatile-rich melts cause channelization of magmatic flow at depths approximately corresponding to the anhydrous solidus temperature. These results are obtained with a novel method to simulate the thermochemical evolution of the upper mantle in the presence of volatiles. The method uses a thermodynamically consistent framework for reactive, disequilibrium, multi-component melting. It is coupled with a system of equations representing conservation of mass, momentum, and energy for a partially molten grain aggregate. Application of this method in two-phase, three-component upwelling-column models demonstrates that it reproduces leading-order features of hydrated and carbonated peridotite melting; in particular, it captures the production of low-degree, volatile-rich melt at depths far below the volatile-free solidus. The models predict that segregation of volatile-rich, deep melts promotes a reactive channelling instability that creates fast and chemically isolated pathways of melt extraction. Reactive channelling occurs where volatile-rich melts flux the base of the silicate melting region, enhancing dissolution of fusible components from the ambient mantle. We find this effect to be similarly expressed for models of both hydrated and carbonated mantle melting. These findings indicate that despite their small concentrations, water and carbon dioxide have an important control on the extent and style of magma genesis, as well as on the dynamics of melt transport.

Tectonic microplates in a wax model of sea-floor spreading

Rotating, growing microplates are observed in a wax analogue model of sea-floor spreading. Wax microplates are kinematically similar to sea-floor tectonic microplates in terms of spreading rate and growth rate. Furthermore, their spiral pseudofault geometry is quantitatively consistent with Schoutens oceanic microplate model. These results suggest that Schoutens edge-driven microplate model captures the kinematics of tectonic microplate evolution on Earth. Based on the wax observations, a theory for the nucleation of overlapping spreading centres, the precursors of tectonic microplates, is developed.

A semi‐Lagrangian Crank‐Nicolson algorithm for the numerical solution of advection‐diffusion problems

We present a hybrid method for the numerical solution of advection-diffusion problems that combines two standard algorithms: semi-Lagrangian schemes for hyperbolic advection-reaction problems and Crank-Nicolson schemes for purely diffusive problems. We show that the hybrid scheme is identical to the two end-member schemes in the limit of infinite and zero Peclet number and remains accurate over a wide range of Peclet numbers. This scheme does not have a CFL stability criterion allowing the choice of time step to be decoupled from the spatial resolution. We compare numerical results with an analytic solution and test both an operator split version of our method and a combined version that solves advection and diffusion simultaneously. We also compare results of simple explicit and implicit numerical schemes and show that the semi-Lagrangian Crank-Nicolson (SLCN) scheme is both faster and more accurate on the same problem. We then apply the combined SLCN scheme to a more geologically relevant benchmark for calculating the thermal structure of a subduction zone. This problem demonstrates that the SLCN scheme can remain stable and accurate at large Courant numbers even in flows with highly curved streamlines. Finally, we introduce a variable order interpolation scheme for the semi-Lagrangian schemes that reduces interpolation artifacts for sharp fronts without introducing numerical diffusion.

The energetics of melting fertile heterogeneities within the depleted mantle

To explore the consequences of mantle heterogeneity for primary melt production, we develop a mathematical model of energy conservation for an upwelling, melting body of recycled oceanic crust embedded in the depleted upper mantle. We consider the end‐member geometric cases of spherical blobs and tabular veins. The model predicts that thermal diffusion into the heterogeneity can cause a factor‐of‐ two increase in the degree of melting for bodies with minimum dimension smaller than ∼1 km, yielding melt fractions between 50 and 80%. The role of diffusion is quantified by an appropriately defined Peclet number, which represents the balance of diffusion‐driven and adiabatic melting. At intermediate Peclet number, we show that melting a heterogeneity can cool the ambient mantle by up to ∼20 K (spherical) or ∼60 K (tabular) within a distance of two times the characteristic size of the body. At small Peclet number, where heterogeneities are expected to be in thermal equilibrium with the ambient mantle, we calculate the energetic effect of pyroxenite melting on the surrounding peridotite; we find that each 5% of recycled oceanic crust diminishes the peridotite degree of melting by 1–2%. Injection of the magma from highly molten bodies of recycled oceanic crust into a melting region of depleted upper mantle may nucleate reactive‐dissolution channels that remain chemically isolated from the surrounding peridotite.