![]()
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 Re, 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...
