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from Part I - Introductory Material
Published online by Cambridge University Press: 26 October 2017
This monograph is a survey of – and introduction to – the ways in which the atmosphere, oceans and various parts of the so-called solid Earth can deform and move. The most common type of deformation is a wave and the most common type of motion is flow. Our goals are to quantify and understand these deformations and motions. In order to achieve these goals, we will need to develop an appropriate set of procedures and tools.
An appropriate procedure is mathematical modeling, which consists of three steps:
• Formulation of a mathematical model of a physical system. This is the crucial element in the procedure. If the model does not mimic the relevant physical processes, it will fail to yield useful results.
• Solution of the mathematical problem. This is the part stressed in most mathematics courses, but in fact is the most routine aspect of the procedure.
• Interpretation of the solution. This is the payoff; solution of a well-conceived model should yield new physical insight and make testable predictions.
We won't be following these steps formally, instead they will come naturally as we investigate each type of wave and flow.
Our mathematical models will treat the physical system (that is, the portion of Earth under consideration) as a continuous body, using the mathematical apparatus of continuum mechanics. The concept of a continuous body is discussed in § 1.2. Continuum mechanics consists of three fundamental elements:
• Kinematics describes how bodies can move and deform. For example, it limits the types of deformation which leave a body competent (i.e., unfractured).
• Dynamics explains why bodies move and deform. They do so in response to external and internal forces, within the context of Newton's second law.
• Rheology quantifies the kinematic response to dynamic forces, taking into account the material properties of the body under investigation. Two identically shaped bodies, composed of water and of steel, respond quite differently to the same applied forces. Rheology encompasses the change of volume of a body induced by a change in pressure (compressive rheology) and the change of shape caused by a change in deviatoric stresses (shear rheology).
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