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31496 results in Solid Earth Geophysics

Page 92 of 1575

  • 6 - Shear Rheology

  • from Part II - Kinematics, Dynamics and Rheology
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 54-66

  • Summary

    Shear rheology, commonly referred to simply as rheology, is the scientific study of the deformational response (strain and rate of strain) of a body to an applied force (stress), often called the stress–strain relation, expressed symbolically as f (S,E, E) = ∅. The response depends principally on the composition of the body, although the temperature and pressure can be important. (For example, the rheology of butter depends rather strongly on the temperature.) If the stress is independent of the rate of strain, the body is elastic and the symbolic relation simplifies to f (S,E) = ∅. Most elastic materials deform only slightly in response to an applied stress and the strain response to stress is adequately described by a linear relationship, obeying Hooke's law; this relationship is developed in § 6.1.

    If the stress is independent of the strain, the body is fluid and the symbolic relation simplifies to f (S,E)=∅. If this relationship is linear, the fluid is said to be Newtonian. The two most commonly encountered fluids, liquid water and air, are Newtonian. However, for many geological materials the relation between stress and flow is nonlinear and the material is non-Newtonian. In general, there are two types of nonlinear rheology: shear thinning and shear thickening. A shear-thinning fluid deforms more readily as the shearing force is increased. Familiar examples of such material are latex paint and quicksand. On the other hand, a shear thickening fluid becomes more resistant to flow as the shearing stress is increased. Many geological materials, including ice (in glaciers), lavas and Earth's silicate mantle behave as shear-thinning fluids. This behavior is investigated in § 6.2.

    The classification of materials as either elastic or fluid is simplistic, and somewhat misleading. For viscoelastic materials the stress depends on both strain and rate of strain. In reality a material may exhibit elastic, fluid or even more complex behavior depending on its thermodynamic state, the magnitude of the stresses applied and the time scale of interest. Most solid materials, as they are heated toward their melting point, develop more fluid-like behavior.1 Many materials (such as emulsions likemayonnaise and whipped cream) behave elastically when the applied stress is small, but change their rheological character to fluid-like or viscoelastic at higher stresses.

  • Geology, biostratigraphy and carbon isotope chemostratigraphy of the Palaeogene fossil-bearing Dakhla sections, southwestern Moroccan Sahara

  • MOULOUD BENAMMI, SYLVAIN ADNET, LAURENT MARIVAUX, JOHAN YANS, CORENTIN NOIRET, RODOLPHE TABUCE, JÉRÔME SURAULT, IMAD EL KATI, SÉBASTIEN ENAULT, LAHSSEN BAIDDER, OMAR SADDIQI, MOHAMED BENAMMI
  • Journal:
    Geological Magazine / Volume 156 / Issue 1 / January 2019
    Published online by Cambridge University Press:
    26 October 2017, pp. 117-132

  • New Palaeogene vertebrate localities were recently reported in the southern Dakhla area (southwestern Morocco). The Eocene sediment strata crops out on cliffs along the Atlantic Ocean coast. Vertebrate remains come from five conglomeratic sandstone beds and are principally represented by isolated teeth belonging to micromammals, selachians and bony fishes, a proboscidean assigned to ?Numidotherium sp. and many remains of archaeocete whales (Basilosauridae). From fieldwork five lithostratigraphic sections were described, essentially based on the lithological characteristic of sediments. Despite the lateral variations of facies, correlations between these five sections were possible on the basis of fossil-bearing beds (A1, B1, B2, C1 and C2) and five lithological units were identified. The lower part of the section consists of rhythmically bedded, chert-rich marine siltstones and marls with thin black phosphorite with organic matter at the base. The overlying units include coarse-grained to microconglomeratic sandstones interbedded with silts, indicating deposition in a shallow-marine environment with fluvial influence. The natural remanence magnetization of a total of 50 samples was measured; the intensity of most of the samples is too weak however, before or after the first step of demagnetization. The palaeomagnetic data from the samples are very unstable, except for eight from three similar sandstone levels which show a normal polarity. Matched with biostratigraphic data on rodents, primates, the selachian, sirenian and cetacean faunas, the new carbon isotope chemostratigraphy on organics (1) refines the age of the uppermost C2 fossil-bearing bed to earliest Oligocene time and (2) confirms the Priabonian age of the B1 to C1 levels.

  • 5 - Some Thermodynamics

  • from Part II - Kinematics, Dynamics and Rheology
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 48-53

  • Summary

    We have investigated how a body may move and deform; this is kinematics, codified in the equation of conservation of mass. We also have investigated the distribution of stresses within a body which may cause it to deform; this is dynamics, codified in the equation of conservation of momentum. Versions of these conservation equations for elastic and fluid bodies are presented in § 4.7. We now turn our attention to the manner in which a body responds to the stresses; this is rheology. Commonly, rheology is taken to mean a change of body shape (called shear rheology), but it also encompasses the change of volume or density (without change of shape) induced by a change of pressure (called compression rheology). This latter response is determined by the equation of state, giving the density ρ in terms of appropriate independent thermodynamic variables. Generally, the equation of state is presented in differential form (as in § 5.2), but for an ideal gas, it is a simple algebraic equation.

    As explained in Appendix E.5, in a system composed of one constituent the density is a function of two independent thermodynamic variables, with some freedom in the choice of those variables. Two useful choices of independent variables are temperature T and pressure p, or pressure and specific entropy s. It should be noted that the density and pressure are unique variables, in that they play leading roles in both thermodynamics and dynamics. The differential form of the equation of state for density is presented in § 5.2, following a discussion in the next section of some new adjectives that we encounter while delving into the dynamics and thermodynamics of geophysical fluids. The thermodynamic presentation in this chapter is rather abbreviated; for more detail, refer to Appendix E.

    New Adjectives

    In our study of geophysical fluids, we will encounter a number of new adjectives. It may be helpful to define these clearly at this point, before plunging into the dynamic and thermodynamic details. These adjectives include:

  • 13 - Nonlinear Shallow-Water Waves

  • from Part III - Waves in Non-Rotating Fluids
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 146-154

  • Summary

    So far we have studied linear deep-water and shallow-water waves. Now we shall turn our attention to nonlinear shallow-water waves, particularly solitary waves. We begin in the following section by re-formulating the shallow-water wave equations to include effects of nonlinearity and dispersion that were ignored in the previous chapter. Then in § 13.2 these equations are scaled and non-dimensionlized. This procedure introduces two dimensionless parameters representing nonlinearity and dispersion. Four limiting cases are identified and studied in the follow four sections, the most interesting of which is the nonlinear dispersive case investigated in § 13.6, involving solitary waves.

    Re-Formulation

    In order to develop the equations governing nonlinear shallow-water waves, let's begin with the full nonlinear formulation for surface-water waves in a body of water having depth D. The governing equation, valid for −D < z < h remains Laplace's equation:

    where ϕ is the velocity potential. The nonlinear kinematic and dynamic conditions at the free surface are

    The bottom boundary condition is simply

    The surface boundary conditions are a pair of coupled nonlinear partial differential equations, applied at an unknown position z = h(x, t). Nonlinear equations are even more difficult to solve than linear equations with variable coefficients, so we have our work cut out for us in attempting to analyze and solve these equations. This provides us with a chance to use some of the tools of the trade, including

  • • scaling the variables;

  • • using dimensional analysis to determine the number of dimensionless parameters;

  • • non-dimensionalizing the variables and equations;

  • • introducing a small parameter;

  • • linearizing the problem based on the small parameter; and

  • • streamlining the notation.

    • Scaling and Non-Dimensionalization

    Let's begin to use our tools by scaling the dependent and independent variables h, ϕ, x, z and t. To begin with, it is obvious that we should scale x with the length of the wave _, z with the water depth D and h with the maximum wave height, hm.

  • 26 - Atmospheric Flows

  • from Part VI - Flows in Rotating Fluids
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 273-276

  • Summary

    The predominant flow within the atmosphere is the general circulation, described in § 26.1. An associated flow, the jet stream, whose existence relies on the thermal wind (considered in § 26.2) is investigated in § 26.3. These flows are powered by the “workhorse” of atmospheric flow: the cumulonimbus rain cloud, which has been investigated in § 24.3.

    General Circulation

    If the Earth were not rotating, the general circulation would consist of a Hadley cell in each hemisphere, with rising motion near the equator, descending motion near the poles, poleward motion at high elevation and equatorward motion near the surface. However, due to the action of the Coriolis force, the general circulation consists of three cells in each hemisphere, a Hadley cell near the equator, a mid-latitude Ferrel cell and the aptly named polar cell at high latitude, as illustrated in Figure 26.1 for the northern hemisphere. The flow in the southern hemisphere is the mirror image, reflected about the equator.

    Circulation of the atmosphere is driven primarily by convective motions in the tropics. Thesemotions drive a circulation of roughly 5 to 8×1010 kg·s−1 in each of the Hadley cells (Liu et al., 2012) and these motions convey poleward a heat flow of roughly 4 PW. The strength of circulation in these cells has been increasing in the past 150 years, possibly due to global warming. The trade winds within the Hadley cells on either side of the equator blow toward the equator and tend to concentrate convective motions in a band close to the equator called the inter-tropical convergence zone (ITCZ) by meteorologists and the doldrums by sailors. Descending motions at the boundary between the Hadley and Ferrel cells produce a band of weak surface winds called the horse latitudes by sailors. These descending motions inhibit convection, so that rainfall is low near the horse latitudes.Most of the world's deserts occur near this latitude, both north and south of the equator.

    The moist adiabatic lapse rate is smaller in magnitude than the dry adiabatic lapse rate.

  • 25 - Ekman Layers

  • from Part VI - Flows in Rotating Fluids
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 253-272

  • Summary

    An Ekman layer is a viscous boundary layer that occurs at a boundary of a large body of rotating fluid: at the bottom of the oceans and atmosphere and at the top of the oceans. This layer is an agent that communicates information regarding velocity (or stress) at the boundary to the fluid outside the layer. The communication is accomplished by means of an Ekman pumping: a vertical flow into (or from) the boundary layer. This vertical flow is determined by integration of the continuity equation, once the horizontal velocity within the layer has been determined by solving the momentum equation.

    Ekman layers are encountered

  • • at the bottom or top of a fluid body;

  • • in either hemisphere; and

  • • adjoining a solid or fluid boundary,

    • giving us eight physical cases to consider. In the following analysis we will keep track of the position of the Ekman layer relative to the boundary and equator is categorized by the factor s which has values 1 or −1, as follows:

  • • s = 1 for bottom layers in the northern hemisphere;

  • • s=−1 for bottom layers in the southern hemisphere;

  • • s=−1 for top layers in the northern hemisphere; and

  • • s = 1 for top layers in the southern hemisphere.

    • There is yet another categorization of Ekman layers, dealing with the nature of the model employed. We will explore two models of Ekman layer flow and structure – having either constant or variable turbulent kinematic viscosity – and compare the results of these two models.

    The equations governing the Ekman layer are presented and solved in § 25.1 and the horizontal Ekman transport is investigated in § 25.2 for both models. Then useful relations between the transport and basal velocity gradient are developed in § 25.3 and Ekman pumping is investigated in § 25.4. Finally, the Ekman-layer theory is applied to the atmosphere and oceans in § 25.5.

  • Part VI - Flows in Rotating Fluids

  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 251-252

  • Summary

    In Part IV we investigated a number of types of waves in the atmosphere and oceans that are affected by rotation. We complement those studies in the present part, by investigating a number of flows that are influenced by rotation. We begin this investigation in the following chapter by investigating the Ekman boundary layer in some detail. This is followed in Chapter 26, with investigation of a number of large-scale flows in the atmosphere, including the general circulation, thermal wind and jet stream. Similarly in Chapter 27 we investigate oceanic currents, including Sverdrup currents, boundary currents and thermohaline circulation. This part concludes with a survey of various atmospheric vortices in Chapter 28, with focus on tornadoes and hurricanes.

Geophysical Waves and Flows
  • Theory and Applications in the Atmosphere, Hydrosphere and Geosphere
  • David E. Loper
  • Published online:
    26 October 2017
    Print publication:
    26 October 2017
  • Waves and flows are pervasive on and within Earth. This book presents a unified physical and mathematical approach to waves and flows in the atmosphere, oceans, rivers, volcanoes and the mantle, emphasizing the common physical principles and mathematical methods that apply to a variety of phenomena and disciplines. It is organized into seven parts: introductory material; kinematics, dynamics and rheology; waves in non-rotating fluids; waves in rotating fluids; non-rotating flows; rotating flows; and silicate flows. The chapters are supplemented by 47 'fundaments', containing knowledge that is fundamental to the material presented in the main text, organized into seven appendices: mathematics; dimensions and units; kinematics; dynamics; thermodynamics; waves; and flows. This book is an ideal reference for graduate students and researchers seeking an introduction to the mathematics of waves and flows in the Earth system, and will serve as a supplementary textbook for a number of courses in geophysical fluid dynamics.

  • Part VII - Silicate Flows

  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 315-316

  • Summary

    We now turn our attention to flows in silicates, including mantle convection, volcanic eruptions and sheet flow of lava on the surface. Like flows in the atmosphere and oceans, the star of these flows is the Navier–Stokes equation, with the continuity equation remaining a co-star, but we now have a much different cast of supporting characters. Rotation, inertia and turbulence, which affect most atmospheric and oceanic flows, are completely negligible in the force balances for silicates. This is good news, because much of the difficulty in analyzing those flows stems from the nonlinear inertia terms; on the face of it the laminar, linearized Navier–Stokes equation would seem to be much easier to solve. However, two new characters that intrude on this idyllic scene are

  • • non-Newtonian creep; and

  • • the strong dependence of creep rate on temperature.

    • Each of these brings nonlinearity into the formulation; the first introduces a velocity nonlinearity while second couples the Navier–Stokes and energy equations. The energy equation contains, among other factors, a nonlinear advective term, presenting us once again with a set of challenging equations to analyze and solve.

    The set of equations governing silicate flows is summarized in the following chapter. Following this, the equations are applied to two rather disparate flow problems: mantle convection and volcanic flows. Earth's mantle convects in order to cool both itself and core. The mantle is cooled by the introduction of cold lithospheric slabs, as explained in Chapter 31, while heat from the core is conveyed upward by discrete plumes, as explained in Chapter 32. Changing gears, Chapter 33 presents an overview of volcanic flows. Then flow in a volcanic conduit is briefly investigated in Chapter 34 and lava sheet flow on the surface is investigated in Chapter 35.

  • 21 - Unsteady Channel Flow: Hydraulic Shock Waves

  • from Part V - Non-Rotating Flows
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 209-217

  • Summary

    The principal aim of this chapter is to determine the transition from super-critical to sub-critical channel flow. While this transition could occur smoothly, as depicted in Figure 20.2 (with the flow arrow reversed), typically the transition is in the form of a hydraulic shock wave that is abrupt (that is, occurring over a small downstream interval) and spontaneous (that is, occurring without any prompting). Hydraulic shock waves are distinct from compressive shock waves which rely on the compressibility of water for the restoring force and which travel far faster than hydraulic shocks. Hydraulic shocks are impelled into motion by the greater water depth on the lee side of the shock. Shallow-water hydraulic shocks have been given differing names depending on the circumstance of their occurrence. Typically the name hydraulic jump is reserved for a shock that is stationary with respect to an observer standing on the ground. Hydraulic jumps commonly occur below a dam when water is released from the base of the reservoir by lifting a sluice gate slightly. Close to the gate the water flows rapidly, but then the depth increases abruptly at a fixed distance downstream from the gate as the flow abruptly transitions from super-critical to sub-critical. A hydraulic shock moving up a river due to the change in tide is called a bore or surge wave. Bores form in river channels during rising tide, most notably in the Amazon and Orinoco in South America that are reported to have bores as high as 4 m. A hydraulic shock having no water in front is called a flash flood. A flash flood wave moves at the mean speed of the water behind the wave.

    Hydraulic shocks often are unsteady when viewed by an observer standing beside the channel, so we must now consider unsteady (time dependent) flows. The details of this transition are complicated, involving turbulent flow and degradation of mechanical energy within a localized downstream interval. In this chapter we will gloss over these details regarding the structure of the shock and investigate the kinematic and dynamic constraints on the uniform flow on either side of a shock. The specific volume flow, q, is conserved across the hydraulic shock.

Page 92 of 1575