The mechanism of deep focus earthquakes has been examined by numerical and linear analysis of shear instability in subducting slabs. We assume subducting slabs deform such that the spatially averaged strain rate becomes time independent. Furthermore, we assume that rheology depends on temperature in the manner of an activation process. Using this model, we investigate quantitatively the conditions under which shear instability takes place, i.e., the deformation in the material concentrates in a thin layer whose temperature increases explosively until melting occurs. First, we study instability in a material whose rheological properties are spatially homogeneous. Shear instability takes place in a homogeneous material when exceeds a certain critical level ε˙c , which depends strongly on the stress in the slabs but does not depend on the detail of the creep law (the value of the activation energy). Next, we study instability in a material that contains shear zones similar to those often observed by field geologists, i.e., a thin layer made of material with lower vicosity than the surrounding region. We find that shear zones trigger shear instability even for the case of provided that G, the ratio of characteristic time of thermal diffusion to that of shear heating in shear zones, exceeds 1, and that f, the ratio of stiffness of elastic deformation in the surrounding region to that of ductile deformation in shear zones, becomes lower than 1. Applying this condition to subducting slabs with shear zones, we find that shear zones trigger shear instability even when the average strain rate is lower than by an order of magnitude. From these results, and from an estimate of in subducting slabs, we conclude that shear instability is a mechanism likely to induce deep focus earthquakes.

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[1] The dynamics of eruption clouds in explosive volcanic eruptions are governed by entrainment of ambient air into eruption clouds by turbulent mixing. We develop a new numerical pseudo gas model of an eruption cloud by employing three-dimensional coordinates, a third-order accuracy scheme, and a fine grid size in order to investigate the behavior of entrainment due to turbulent mixing. The quantitative features of entrainment are measured by a proportionality constant relating the inflow velocity at the edge of the flow to the average vertical velocity (i.e., the entrainment coefficient). Our model has successfully reproduced the quantitative features of entrainment observed in the laboratory experiments as well as fundamental features of the dynamics of eruption clouds, such as the generation of eruption columns and/or pyroclastic flows. The value of the entrainment coefficient for eruption clouds is estimated from the column height and critical condition for column collapse by comparing results of our model with those of previous one-dimensional models. It is suggested that the value of the entrainment coefficient for an eruption cloud is approximately constant, although the value estimated from the critical condition for column collapse (k $ 0.07) is slightly smaller than that based on the column height (k $ 0.1). This difference reflects the vertical change of flow structure in the eruption cloud. The eruption cloud in the upper region exhibits a meandering instability, which leads to efficient mixing, whereas the cloud near the vent maintains a concentric structure with an inner dense core surrounded by an outer shear region. Our model is consistent with previous one-dimensional models for steady eruption clouds supported by the laboratory experiments, and it is also applicable to unsteady and transient features of actual eruption clouds.Citation: Suzuki, Y. J., T. Koyaguchi, M. Ogawa, and I. Hachisu (2005), A numerical study of turbulent mixing in eruption clouds using a three-dimensional fluid dynamics model,

Numerical calculations are presented for the steady three-dimensional structure of thermal convection of a fluid with strongly temperature-dependent viscosity in a bottom-heated rectangular box. Viscosity is assumed to depend on temperature T as exp (− ET), where E is a constant; viscosity variations across the box r (= exp (E)) as large as 105 are considered. A stagnant layer or lid of highly viscous fluid develops in the uppermost coldest part of the top cold thermal boundary layer when r > rc1, where r = rc1 ≡ 1.18 × 103Rt0.309 and Rt is the Rayleigh number based on the viscosity at the top boundary. Three-dimensional convection occurs in a rectangular pattern beneath this stagnant lid. The planform consists of hot upwelling plumes at or near the centre of a rectangle, sheets of cold sinking fluid on the four sides, and cold sinking plume concentrations immersed in the sheets. A stagnant lid does not develop, i.e. convection involves all of the fluid in the box when r < rc1. The whole-layer mode of convection occurs in a three-dimensional bimodal pattern when r > rc2 = 3.84 × 106Rt−1.35. The planform of the convection is rectangular with the coldest parts of the sinking fluid and the hottest part of the upwelling fluid occurring as plumes at the four corners and at the centre of the rectangle, respectively. Both hot uprising plumes and cold sinking plumes have sheet-like extensions, which become more well-developed as r increases. The whole-layer mode of convection occurs as two-dimensional rolls when r < min (rc1, rc2). The Nusselt number Nu depends on the viscosity at the top surface more strongly in the regime of whole-layer convection than in the regime of stagnant-lid convection. In the whole-layer convective regime, Nu depends more strongly on the viscosity at the top surface than on the viscosity at the bottom boundary.

[1] A series of numerical models are presented for two-dimensional thermal convection in a fluid with viscosity that nonlinearly depends on ''degree of damage'' w as well as temperature and pressure; w increases and viscosity decreases with time when the convection induces strong viscous dissipation. The convecting fluid recovers from the damage with a characteristic time that depends on temperature. The w dependence of viscosity induces a stress-viscosity relationship that consists of two branches, the ''intact branch'' at low stress and the ''damaged branch'' at high stress, with a hysteresis between the two branches; w is small (large) and viscosity is high (low) on the intact (damaged) branch. The hysteresis lets the viscosity depend on stress history. The dependence of viscosity on stress history due to the hysteresis induces a regime where the numerically modeled lithosphere along the surface boundary is fragmented into smaller pieces that are on the intact branch and rigidly move. The pieces of lithosphere are separated by narrow mechanically weak zones that are on the damaged branch. Each piece of lithosphere is further fragmented and mechanically weak zones are newly formed only when the piece is tapped by particularly strong hot uprising plumes. This ''plate-like'' regime is the only regime, among the convective regimes searched here, where each piece of lithosphere rigidly moves and still is not ruptured by the forces that drive the piece. The Earth's mantle is suggested to be on the plate-like regime. Citation: Ogawa, M., Plate-like regime of a numerically modeled thermal convection in a fluid with temperature-, pressure-, and stress-history-dependent viscosity,