Fluid and Solid Mechanics

Shaun Bullett, Tom Fearn;Frank Smith

• Publisher: WSPC (EUROPE)
• Year: 2016
• ISBN:
  9781786340283
• Topic: Physical Sciences
• Subtopic: Fluid Mechanics

This book leads readers from a basic foundation to an advanced-level understanding of fluid and solid mechanics. Perfect for graduate or PhD mathematical-science students looking for help in understanding the fundamentals of the topic, it also explores more specific areas such as multi-deck theory, time-mean turbulent shear flows, non-linear free surface flows, and internal fluid dynamics.

Fluid and Solid Mechanics is the second volume of the LTCC Advanced Mathematics Series. This series is the first to provide advanced introductions to mathematical science topics to advanced students of mathematics. Edited by the three joint heads of the London Taught Course Centre for PhD Students in the Mathematical Sciences (LTCC), each book supports readers in broadening their mathematical knowledge outside of their immediate research disciplines while also covering specialized key areas.

Contents:

• Introductory Geophysical Fluid Dynamics (Michael Davey)
• Multiple Deck Theory (S N Timoshin)
• Time-Mean Turbulent Shear Flows: Classical Modelling — Asymptotic Analysis — New Perspectives (Bernhard Scheichl)
• Nonlinear Free Surface Flows with Gravity and Surface Tension (J-M Vanden-Broeck)
• Internal Fluid Dynamics (Frank T Smith)
• Fundamentals of Physiological Solid Mechanics (N C Ovenden and C L Walsh)

Readership: Researchers, graduate or PhD mathematical-science students who require a reference book that covers fluid dynamics and solid mechanics.
Pure Mathematics;Applied Mathematics;Mathematical Sciences;Techniques;Algebra;Logic;Combinatorics;Fluid Dynamics;Solid Mechanics Key Features:

• Each chapter is written by a leading lecturer in the field
• Concise and versatile
• Can be used as a masters level teaching support or a reference handbook for researchers

Chapter 1

Introductory Geophysical Fluid Dynamics

Michael Davey

Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Centre for Mathematical Sciences,
Wilberforce Road, Cambridge CB3 0WA, UK
[email protected]

This chapter concerns mathematical modelling of large-scale fluid flows relative to a rotating frame of reference, for which the effects of rotation are dominant and to leading order there is a balance of horizontal pressure gradients and Coriolis forces. The principal application is to oceanic and atmospheric flows with horizontal scales of tens of kilometres or more, and timescales of days or more. A fundamental equation in the dynamics of such flows is that for quasigeostrophic potential vorticity, and this is derived in the first part of the chapter, with stratification effects included in the form of layers with constant density within each layer. Large-scale wave-like behaviour is supported in the form of Rossby waves, and some basic properties of these waves are presented. Simplified conceptual quasigeostrophic models provide understanding of dynamical processes, and two examples are described: ocean spin-up and multiple equilibria.

1.Introduction

Mathematical representation of large-scale atmospheric and oceanic flows has great practical importance as it provides the basis for the dynamical numerical models used for making weather and climate outlooks for hours to decades ahead. The full equations of fluid motion are too complex to use for this purpose, but mathematical theory provides the foundation for approximations that represent the scales and phenomena of interest and allow efficient numerical computation. Even with these approximations, atmospheric and oceanic flows contain processes and interactions on a wide range of space and time scales. Mathematical models can further be used to focus on particular processes and investigate their behaviour and roles.

This chapter contains a subset of a course intended for graduates who are familar with the basics of fluid mechanics, such as the Navier–Stokes equations and wave-like behaviour such as gravity waves, but have not encountered geophysical fluid dynamics. A brief explanation of the governing equations for quasigeostrophic flow is provided, without rigorous justification for the various standard approximations employed.

Two examples are provided of conceptual models based on the quasigeostrophic potential vorticity equations. One is a model of mid-latitude wind-driven ocean circulation. The classic steady case demonstrates how intense currents such as the Gulf Stream occur near western boundaries, while the time-dependent part illustrates how Rossby waves influence ocean circulation and how in a stratified ocean they provide oceans with a long-term “memory”. The second example demonstrates how the interaction of Rossby waves, topographic drag and mean flow may create multiple stable states, relevant in particular to “blocked” flow regimes in the atmosphere.

There are many good textbooks on this subject. More detailed and rigorous derivations of sets of equations relevant to geophysical fluid dynamics, with applications, may be found in books by Gill,¹ Pedlosky² and Vallis³ for example.

2.Governing Equations

For flows relative to a rotating frame of reference, the Navier–Stokes equations have the form

where u is the velocity vector, p is pressure, ρ is density, D/Dt indicates a derivative following the motion and ν is a viscosity coefficient. For planet Earth, the rotation vector Ω has magnitude Ω = 2π radians per day and direction outward from the North Pole. Earth can be regarded as a sphere of radius R_e, with the atmospheric and oceanic flows in thin layers near that radius, with large horizontal scale compared to the depth in each medium.

With flows in mid-latitude regions in mind, choose a coordinate system that is centred on some latitude θ₀. For simpli...