This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

Episodic continental growth and supercontinents : a mantle avalanche connection?

The study of noble gases in oceanic basalts is central to understanding chemical heterogeneity of the Earth’s mantle and origin of the atmosphere. In the terrestrial environment the abundances of noble gases are quite low because they were excluded from solid materials during planetary formation in the inner solar system. This low background inventory helps to make the noble gases excellent tracers of mantle reservoirs. In this context, mid-ocean ridge and ocean island basalts provide valuable windows into the Earth’s mantle. These oceanic basalts are not prone to the degree of contamination often observed in continental lavas that results from their passage through thick continental lithosphere and crust. Mid-ocean ridge basalts (MORBs) form by partial melting as the ascending mantle beneath spreading ridges reaches its solidus temperature, and MORBs are generally accepted to represent a broad sampling of the convecting upper mantle. Ocean island basalts (OIBs) represent melting ‘anomalies’ that are generally related to mantle upwelling. The extent to which ocean islands are derived from a thermal boundary layer in the deep mantle (e.g., as a mantle plume) or from chemical heterogeneities embedded within the mantle convective flow (e.g., as a mantle ‘blob’) has been debated for decades, and is not currently resolved. The isotope compositions of noble gases in oceanic basalts bear significantly on such debates over the chemical structure of the mantle. When oceanic basalts erupt as submarine lavas, their quenched rims of glass may contain high volatile abundances (especially when they are deeply erupted under elevated hydrostatic pressure), providing the best available opportunity for precisely characterizing the noble gas composition of the Earth’s mantle. In favorable cases, inclusions of melt or fluids trapped within magmatic phenocrysts and mantle xenoliths can also be precisely analyzed for noble gas composition.

Measurable changes in the isotope compositions of noble gases …

Noble Gas Isotope Geochemistry of Mid-Ocean Ridge and Ocean Island Basalts: Characterization of Mantle Source Reservoirs

Because of their distinct chemical signatures, ocean-island and mid-ocean-ridge basalts are traditionally inferred to arise from separate, isolated reservoirs in the Earth's mantle. Such mantle reservoir models, however, typically satisfy geochemical constraints, but not geophysical observations. Here we propose an alternative hypothesis that, rather than being divided into isolated reservoirs, the mantle is filtered at the 410-km-deep discontinuity. We propose that, as the ascending ambient mantle (forced up by the downward flux of subducting slabs) rises out of the high-water-solubility transition zone (between the 660 km and 410 km discontinuities) into the low-solubility upper mantle above 410 km, it undergoes dehydration-induced partial melting that filters out incompatible elements. The filtered, dry and depleted solid phase continues to rise to become the source material for mid-ocean-ridge basalts. The wet, enriched melt residue may be denser than the surrounding solid and accordingly trapped at the 410 km boundary until slab entrainment returns it to the deeper mantle. The filter could be suppressed for both mantle plumes (which therefore generate wetter and more enriched ocean-island basalts) as well as the hotter Archaean mantle (thereby allowing for early production of enriched continental crust). We propose that the transition-zone water-filter model can explain many geochemical observations while avoiding the major pitfalls of invoking isolated mantle reservoirs.

Whole-Mantle Convection and the Transition-Zone Water Filter

The supercontinent cycle: A retrospective essay

Mantle Convection in the Earth and Planets is a comprehensive synthesis of all aspects of mantle convection within the Earth, the terrestrial planets, the Moon, and the Galilean satellites of Jupiter. Mantle convection sets the pace for the evolution of the Earth as a whole. It influences Earth’s topography, gravitational field, geodynamo, climate system, cycles of glaciation, biological evolution, and formation of mineral and hydrocarbon resources. It is the primarymechanism for the transport of heat from the Earth’s deep interior to its surface. Mantle convection is the fundamental cause of plate tectonics, formation and drift of continents, volcanism, earthquakes, and mountain building. This book provides both a connected overview and an in-depth analysis of the relationship between these phenomena and the process of mantle convection. Complex geodynamical processes are explained with simple mathematical models. The book includes up-to-date discussions of the latest research developments that have revolutionized our understanding of the Earth and the planets. These developments include:

/pdf/mantle-convection-in-the-earth-and-planets-3pibpenswm.pdf

Mantle Convection in the Earth and Planets

http://www1.gly.bris.ac.uk/~jmk/PAPERS/tine_etal_epsl04.pdf

Lower-mantle seismic discontinuities and the thermal morphology of subducted slabs

Distinct rigidly moving oceanic and continental plates of finite thickness are incorporated into a two-dimensional numerical model of mantle convection. We investigate upper mantle convection in models having aspect ratios as great as 24 and compare our findings with the results of earlier studies which were limited to aspect ratio 4 models. In addition, we implement models of whole mantle flow by specifying high Rayleigh number convection and thinner nondimensional plates. We are thus able to compare the results of continental collision models which include similarly sized continents in the cases of upper and whole mantle convection. For each case considered we model a pair of identical continents being carried toward a site of plate convergence by underlying counterrotating mantle convection cells. Upon collision, the continents form a motionless, rigid, model supercontinent, while oceanic plate material continues to recycle through the mantle. Following the continental collision, our models of upper mantle convection exhibit a reorganization of the convective planform below the model supercontinent into a smaller wavelength mode which is unable to generate the net stress needed to break apart the continent; alternating compressive and tensile deviatoric stress associated with the small scale flow results in a low integrated stress. In contrast the large scale of whole mantle convection enables flow reversals to produce shear stresses acting in a common direction over extensive areas of the base of a continent, the integrated effect of which is capable of causing continental rifting. The conventional view of the role of thermal blanketing in continental rifting does not apply in the whole mantle convection scenario.

Julian P. Lowman

Papers

Mantle Convection in the Earth and Planets

Lower-mantle seismic discontinuities and the thermal morphology of subducted slabs

Continental collisions in wide aspect ratio and high Rayleigh number two‐dimensional mantle convection models

The influence of mantle internal heating on lithospheric mobility: Implications for super-Earths

Episodic tectonic plate reorganizations driven by mantle convection