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*Leigh H. A semi-analytic, 3-D model for subduction within a Newtonian viscous upper mantle provides a dynamically consistent means of computing viscous stress, trench motion and slab geometry in subduction systems. Steady-state slab dip increases as slab density decreases, especially for very low-density slabs, which dip significantly more steeply than high-density slabs.*

We shall recall our definition of relative inertial reference frames. Choose a coordinate system Figure We shall now show that the relative velocity between the two particles is independent of the choice of reference frame providing that the reference frames are relatively inertial. For a two-particle interaction, the relative velocity between the two vectors is independent of the choice of relatively inertial reference frames. Figure

## 15.2: Reference Frames and Relative Velocities

Leigh H. A semi-analytic, 3-D model for subduction within a Newtonian viscous upper mantle provides a dynamically consistent means of computing viscous stress, trench motion and slab geometry in subduction systems. Steady-state slab dip increases as slab density decreases, especially for very low-density slabs, which dip significantly more steeply than high-density slabs.

The horizontal velocity at the top of the lower mantle, measured relative to the foreland, has a very large effect on trench migration rates, rivalling or even exceeding that of slab buoyancy. Slab width, parallel to the trench, also has a significant effect on trench migration rates due to the viscous pressure of toroidal flow around the slab. The stiffness of the subducting lithosphere does not exert a significant effect on trench migration rates or slab geometries for rigidities compatible with oceanic lithosphere.

Large, unexpected effects on trench migration rates and slab geometry are exerted by the structure and density of the frontal prism and overriding plate, indicating that local geology can exert important constraints on slab dynamics. During non-steady-state subduction, rates of trench migration respond rapidly as variably buoyant lithosphere penetrates into the asthenosphere.

Subduction of variable-buoyancy lithosphere is accompanied by changes in slab dip with depth and through time. Subduction systems embody the local interplay between slab buoyancy and mantle viscosity, embedded within the global convective system e. Within this interactive dynamic system, the factors that determine the motion of the trenches are poorly understood.

Griffiths et al. The failure to incorporate self-consistent trench motion into dynamic models for subduction affects the results of subduction-zone models and limits our understanding of the subduction process.

This paper was largely motivated by the desire to understand the evolution of subduction systems in which variable slab densities and variable rates of trench motion appear to play a primary role in the regional tectonics, for example within the Mediterranean region e. In the process of our investigations, we discovered that, although many quantitative analyses of subduction exist, few were framed in a way that could be readily adapted to the subduction of narrow, variable-density slabs exceptions include works by Dvorkin et al.

Eventually, we decided develop an analytical approach that was applicable to steady-state and time-dependent systems, beginning with analysis and quantification of the stresses that act on subducting lithosphere and the way in which these stresses are related to the evolving slab geometry and trench velocity. The dynamic behaviour of subduction systems is governed by the balance between stresses induced by the buoyancy of the slab and by the viscous flow of mantle adjacent to the slab e.

In general, mantle flow related to subduction is 3-D and cannot be readily captured by 2-D flow models e. Funiciello et al. In this paper, we consider the case where the subducting slab may migrate laterally through the surrounding asthenosphere and where subduction is sufficiently far advanced that the slab extends to the base of the upper mantle Fig. Because the lower mantle is too viscous to flow at rates comparable to the upper mantle, slab migration must be accompanied by lateral flow of material around the edges of the slab toroidal flow , as well as by downdip flow parallel to the slab poloidal flow.

This means that the buoyancy stresses arising from the downgoing slab are resisted by a viscous mantle with important components of flow in three-dimensions see model results of Funiciello et al. Schematic slab geometry, showing trench migration white arrows , toroidal flow around slab grey-shaded arrow heads and direction of net mantle flux within viscous wedges above and below the plate black arrowheads.

Stegman et al. One of the difficulties in allowing for self-consistent trench motion is that most finite-element models of convection cannot allow for a truly stress-free upper surface. Instead, a vertical component of velocity, usually zero, is specified at the upper surface of the model.

In practice, this means that trench location must be specified explicitly and that it is difficult to allow for trench migration in a natural, self-consistent manner. Partially for this reason, laboratory experiments of subduction have been used to generate insight into trench migration as a self-consistent feature arising from subduction Bellahsen et al.

There is also much to be learned from simple algorithms that approximate the interactions of a slab with the surrounding mantle and allow self-consistent trench motion. One such 3-D approach was used by Dvorkin et al. A second example is the work of Buiter et al. We use a somewhat similar approach, but one that does not depend a priori on specified forces, slab geometries or rates.

In the following sections we develop a dynamically consistent approach to subduction that involves coordination of several algorithms, or modules, each of which describes the behaviour of one part of the system. One module describes stresses due to the poloidal flow of mantle above the slab, another describes the stresses due to toroidal circulation of mantle around the slab, yet another describes the rheologic behaviour of the slab itself, etc.

While each module is based on an analytic algorithm, the subduction process is simulated by linking all the modules at one time and time stepping through the subduction process. We stress that, if desired, any particular module can be reformulated without changing the overall approach to the problem.

For example, the slab could be modelled using either a viscous or elastic thin-sheet rheology, a different approximation could be used for flow and stresses within the circulating mantle, etc. In the following sections we go through the physics of each element of the subduction problem, beginning with the stresses that act on the subducting lithosphere.

See Table 1 and text for definition of other variables. Slabs with pre-subduction water depths near zero, typical of much continental lithosphere, have a buoyancy of approximately the same magnitude, but opposite sign.

Note that this method of estimating buoyancy refers only to the lithospheric material that is actually subducted. If, for example, sedimentary layers were stripped off the top of the slab prior to subduction, the buoyancy of the subducted slab would be determined by isostatically removing the sediments from the basement and calculating the resultant water depth; such backstripping of sedimentary sequences is a common technique in basin analysis.

Within the frontal prism, we assume that the deformation is negligible, so that total stress within the frontal prism is equal to the lithostatic pressure. Under this assumption, the total normal stress on the upper surface of the slab beneath the frontal prism is equal to the lithostatic pressure and can be computed from Eq. Similarly, the shear stress on the upper surface of the slab beneath the frontal prism is zero.

These assumptions can easily be modified to include an explicit rheology for the frontal prism, so that the stress it applies to the top of the slab will include lithostatic pressure and the stress of deformation. We also assume that the bathymetry of the upper surface of the frontal prism is equal to the initial water depth of the incoming plate so that no bathymetric trench is formed either of these two assumptions can be easily modified, as desired.

The viscous stresses related to subduction can be approximately subdivided into the long wavelength, or background, stresses that are related to large-scale flow of mantle material around the slab, and the highly localized stresses that are related to the flux of material into the narrow portions of the upper and lower mantle wedges Figs 1—3.

This subdivision is useful because the large-scale flow of mantle material around the slab can, at long wavelength, be approximated by 2-D flow in the x — y plane, while the influx of material into the narrow mantle wedges can, away from the edges of the slab, be approximated by 2-D flow in the x — z plane. Hence a difficult 3-D problem can be approximated by two simpler problems in 2-D flow.

The frame of reference is a point on the foreland located far from the subduction boundary, so that the slab moves downwards and laterally at a rate v R. The lower mantle moves in the x -direction with velocity v m and the overriding plate moves with velocity v t.

For simplicity, the velocity of the overriding plate was set equal to the velocity of the trench, so that no deformation occurs within the overriding plate. However, other assumptions can be made, the effects of which are mentioned in the results section of this paper.

Plan view of Hele-Shaw flow field used to approximate viscous pressure of toroidal flow around a slab of half-width a , moving in the horizontal plane at a rate v R. Note that our convention of denoting velocities with respect to the foreland lithosphere is different from the usual convention of computing velocity with respect to the top of the lower mantle or some other reference frame, e. In the wedge of viscous mantle that overlies the slab, the local flow pattern and related viscous stresses will be fundamentally different from the overall toroidal flow pattern, as material is fluxed from thicker to thinner parts of the wedge Fig.

This is particularly the case within the narrow portions of the wedge near the asthenospheric nose where flow is severely constricted by the plates above and below. Viscous stresses in this domain are important not only in understanding rates of trench migration but also for lattice preferred orientation of olivine in the mantle wedge e.

Kaminski et al. We note, however, that the 2-D solution breaks down near the edges of the slab where even local flow is fundamentally 3-D. Building on the relationships between slab velocity, geometry and stress established in the preceding sections and the appendices, we can construct an algorithm for the behaviour of the subducted slab, where the externally applied stresses and the slab buoyancy are united via the rheology of the subducted slab and the surrounding mantle. For a given slab geometry and rate of trench migration, the viscous stresses on the slab will vary with the viscosity of the sublithospheric mantle, and vice versa.

In our model results presented below there is only a narrow range of values for upper mantle viscosity that yield subduction rates consistent with observed rates of subduction around 2—3 10 20 Pa s, Fig. However, there is considerable latitude in our model of how viscosity may vary with depth.

Published estimates of the viscosity of the upper mantle suggest that its lower part may be several to perhaps 10 times more viscous than its upper part e. In general, we use a two-layer viscosity structure with an asthenospheric viscosity of 2. For comparison, we also give some results for a uniform viscosity upper mantle of 3. The viscosity of the lower mantle beginning at km depth is assumed to be very much greater than that of the upper mantle e.

The crust of the overriding plate is These values were chosen because they give a water depth of zero for an isostatically compensated upper plate. We quantify the flexure of the slab in response to bending stresses as that of a thin flexurally competent sheet.

The slab is able to transmit extensional and compressional stresses along its length, consistent with studies that suggest that the slab may be able to support much of its weight within the upper mantle e.

The downdip length of the slab is not allowed to change during bending. Thus slab rigidity has a negligible effect on slab behaviour as compared to the zero-strength case but its inclusion helps to maintain numerical stability in the computations because it prevents kinking of the slab. In the results section we will show that slab rigidity has negligible effect on subduction geometry or rates until it exceeds an effective elastic thickness of 30—40 km, with similar results for a viscous slab.

Note that q n and q s depend on the viscous stress on the top and bottom of the slab, and so are related to the velocity of the slab [e. Appendix C combines the external and gravitational stresses acting on the slab with an equation that includes the bending stresses internal to the slab.

By setting the total horizontal and vertical forces acting on each slab element equal to zero, we derive one equation that links all the external stresses on the slab to the velocity and geometry of the slab eq. Of great importance is where to terminate the narrow end of the viscous wedge above the slab because stresses near the asthenospheric nose become extremely high. However, in our description, there is a natural way to choose the location of the asthenospheric nose using the concept that stresses must be continuous throughout the region modelled.

Thus the total pressure in the asthenospheric nose should be comparable to the total pressure in the adjacent part of the frontal prism. In other words, the pressure immediately in front of the viscous wedge needs to be sufficiently low that viscous asthenospheric material is drawn into the narrow part of the wedge by the lateral pressure gradient. More correctly, the total stress must be continuous, but given uncertainties in the precise configuration of the asthenospheric the probable gradational transition from asthenosphere to lithosphere, assuming a continuous pressure field should be a sufficiently accurate condition for internal consistency of stresses.

At each time step we solve for the location of the front of the asthenospheric nose by finding the slab depth for which eq. In practice, v s and v n are a weighted average of the slab velocities for the previous 0. This damping process prevents oscillation in the results due to overshooting of the correct values of velocity and stress.

We then find the location on the slab where eq. Somewhat more problematic is how to treat the narrow end of the viscous wedge below the subducting slab and the interaction of the subducting slab with the lower mantle. Almost certainly, the subducting slab does not behave as a coherent plate at these depths e. This issue becomes especially important when there is relative motion between the foreland lithosphere and the top of the lower mantle.

First, velocities at the base of the upper mantle e. The motion of the slab will be affected by this flow. The second effect results from the deep slab becoming anchored, or partly anchored, to the lower mantle due to the high viscosity in the lower mantle. The latter presents problems if the slab is not allowed to undergo longitudinal strain and if the slab is still strong enough to act as a stress guide, as in our slab model.

In this case, stress transmission along the slab may interfere with the subduction processes near the surface in a manner that we believe to be physically unrealistic.

In order to avoid having to deal with the interaction of the slab with the high-viscosity lower mantle, we truncate the downgoing slab where the lower surface of the slab has descended to 95 per cent of the total depth of the upper mantle, approximately 30 km above the top of the lower mantle.

Thus, in all the formulations and results in this paper, the slab is not attached to the lower mantle in any way. Slab motion at each time step was derived by solving eq.

## 15.2: Reference Frames and Relative Velocities

Recent groundbreaking work in cognitive linguistics has revealed the semantic complexity of motion metaphors of time and of temporal frames of reference. In most approaches the focus has been on the clause-level metaphorical meaning of expressions, such as Moving Ego We are approaching the end of the year and Moving Time both Ego-centered , as in The end of the year is approaching and field-based , as in Boxing Day follows Christmas Day. The detailed grammatical structure of these metaphorical expressions, on the other hand, has received less attention. Such details include both elements that contribute to the metaphorical meaning and those that have a non-metaphorical temporal function, e. I propose a model for the analysis of metaphorical expressions, building on earlier work in Conceptual Metaphor Theory and the framework of Cognitive Grammar CG. I approach the grammatical structure of metaphorical expressions by analyzing the interplay between veridical and metaphorical systems of expressing temporal relations. I argue that these systems relate to two relevant conceptualizations of time.

In physics , the Coriolis force is an inertial or fictitious force [1] that acts on objects that are in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise or counterclockwise rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the Coriolis effect. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an paper by French scientist Gaspard-Gustave de Coriolis , in connection with the theory of water wheels.

*Kinematic equations relate the variables of motion to one another. In Part 3, vectors are used to solve the problem.*

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A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. While the laws of motion are the same in all inertial frames, in non-inertial frames, they vary from frame to frame depending on the acceleration. In classical mechanics it is often possible to explain the motion of bodies in non-inertial reference frames by introducing additional fictitious forces also called inertial forces, pseudo-forces [4] and d'Alembert forces to Newton's second law. Common examples of this include the Coriolis force and the centrifugal force.

In the rotating coordinate system, B will observe that the point r, which is fixed in inertial space appears to move backwards due to the rotation of B s coordinate.

Kinematic is a subfield of physics, developed in classical mechanics , that describes the motion of points, bodies objects , and systems of bodies groups of objects without considering the forces that cause them to move. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics , not kinematics. For further details, see analytical dynamics.

Стоит ли удивляться, что он находится на грани срыва?. - С учетом обстоятельств, я полагаю, - сказала Сьюзан, - вам все же нужно позвонить директору. Стратмор покачал головой, и капля пота с его лба упала на стол. - Я не хочу никоим образом нарушать покой директора и говорить с ним о кризисе, в то время как он не в состоянии предпринять хоть что-нибудь. Сьюзан понимала, что коммандер прав.

* - Помните, я сказала, что на Нагасаки сбросили плутониевую бомбу.*