Plate generation and two-phase damage theory in a model of mantle convection

The formation of narrow, rapidly deforming plate boundaries separating strong plate interiors are integral components of the generation of plate tectonics from mantle convection. The development of narrow plate boundaries requires the interaction of a non-linear rheology and convection. One such non-linear rheology is two-phase damage theory which employs a non-equilibrium relation between interfacial surface energy, pressure and viscous deformation, thereby forming a theoretical model for void generation. Two-phase damage theory was recently extended to allow for deformational work to increase the fineness (reduce the grain size) of the matrix phase. We present results testing two-phase damage theory in a 2-D convectively driven system where we allow for (1) pure void-generating damage, (2) pure fineness-generating damage and (3) combined void- and fineness-generating damage. Pure void-generating damage is found to be unsuccessful at producing plate-like features. Fineness-generating damage is successful at inducing plate-like behaviour in certain circumstances, including increasing viscosity sensitivity to fineness and certain regimes of damage input and healing rate. Cases with combined void- and fineness-generating damage produce significantly more localization than the end-members due to the apparent increase of deformational work input into fineness generation. The interaction of microcracks and grain size reduction in two-phase damage theory suggests a rheological model for shear localization necessary for the formation of plate tectonic boundaries.     

Variations in planetary convection via the effect of climate on damage

A new model for the generation of plate tectonics suggests an important interaction between a planet's climate and its lithospheric damage behavior; and thus provides a simple explanation for the tectonic difference between Earth and Venus. We propose that high surface temperatures will lead to higher healing rates (e.g. grain growth) in the lithosphere that will act to suppress localization, plate boundary formation, and subduction. This leads to episodic or stagnant lid convection on Venus because of its hotter climate. In contrast, Earth's cooler climate promotes damage and plate boundary formation. The damage rheology presented in this paper attempts to describe the evolution of grain size by allowing for grain reduction via deformational work input and grain growth via surface tension-driven coarsening. We explore the interaction of damage and healing in two-dimensional numerical convection simulations. We also develop a simple “drip-instability” model to test the hypothesis that the competition between damage and healing controls convective and plate tectonic style by modulating episodicity at subduction zones. At small values of damage, f_A, (or large values of healing, k_A) the lithosphere remains strong enough to resist subduction on time scales of billions of years. At intermediate values of f_A and k_A the lithosphere may become mobilized and allow for short bursts of tectonic behavior followed by periods of quiescence. At large (small) values of f_A (k_A ) the fineness is increased so that the viscosity of the plate boundary is reduced to allow for continuous, unimpeded subduction of lithosphere and plate-like deformation. The results suggest the feasibility of our proposed hypothesis that the interplay of climate and damage control the mode of tectonics on a planet.     

Divergent evolution of Earth and Venus: Influence of degassing,tectonics, and magnetic fields

Knowledge of the earliest evolution of Earth and Venus is extremely limited, but it is obvious from their dramatic contrasts today that at some point in their evolution conditions on the two planets diverged. In this paper we develop a geophysical systems box model that simulates the flux of carbon through the mantle, atmosphere, ocean, and seafloor, and the degassing of water from the mantle. Volatile fluxes, including loss to space, are functions of local volatile concentration, degassing efficiency, tectonic plate speed, and magnetic field intensity. Numerical results are presented that demonstrate the equilibration to a steady state carbon cycle, where carbon and water are distributed among mantle, atmosphere, ocean, and crustal reservoirs, similar to present-day Earth. These stable models reach steady state after several hundred million years by maintaining a negative feedback between atmospheric temperature, carbon dioxide weathering, and surface tectonics. At the orbit of Venus, an otherwise similar model evolves to a runaway greenhouse with all volatiles in the atmosphere. The influence of magnetic field intensity on atmospheric escape is demonstrated in Venus models where either a strong magnetic field helps the atmosphere to retain about 60 bars of water vapor after 4.5 Gyr, or the lack of a magnetic field allows for the loss of all atmospheric water to space in about 1 Gyr. The relative influences of plate speed and degassing rate on the weathering rate and greenhouse stability are demonstrated, and a stable to runaway regime diagram is presented. In conclusion, we propose that a stable climate-tectonic-carbon cycle is part of a larger coupled geophysical system where a moderate surface climate provides a stabilizing feedback for maintaining surface tectonics, the thermal cooling of the deep interior, magnetic field generation, and the shielding of the atmosphere over billion year time scales.     

Interpolation with Splines in Tension: A Green''s Function Approach

Interpolation and gridding of data are procedures in the physical sciences and are accomplished typically using an averaging or finite difference scheme on an equidistant grid. Cubic splines are popular because of their smooth appearances; however, these functions can have undesirable oscillations between data points. Adding tension to the spline overcomes this deficiency. Here, we derive a technique for interpolation and gridding in one, two, and three dimensions using Green's functions for splines in tension and examine some of the properties of these functions. For moderate amounts of data, the Green's function technique is superior to conventional finite-difference methods because (1) both data values and directional gradients can be used to constrain the model surface, (2) noise can be suppressed easily by seeking a least-squares fit rather than exact interpolation, and (3) the model can be evaluated at arbitrary locations rather than only on a rectangular grid. We also show that the inclusion of tension greatly improves the stability of the method relative to gridding without tension. Moreover, the one-dimensional situation can be extended easily to handle parametric curve fitting in the plane and in space. Finally, we demonstrate the new method on both synthetic and real data and discuss the merits and drawbacks of the Green's function technique.