Abstract
The purpose of this chapter is to familiarize the reader with the equations governing the motion of atmospheres and the oceans. It introduces the well-known Boussinesq and hydrostatic approximations, and outlines the way the shallow water equations and the quasi-geostrophic equations follow from these approximations. The importance of boundary conditions is also emphasized. Readers who are familiar with these equations, approximations, and conditions may skip this chapter without loss of continuity.
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Notes
- 1.
Velocity is normally referred to as wind in the atmosphere and current in the ocean, seas, and lakes.
- 2.
In the following, bold upright fonts, e.g., \(\mathbf {u}\), denote vectors, while bold special italic fonts, e.g., \(\mathscr {U}\), denote tensors.
- 3.
R \(=\) 287.04Â JÂ kg\(^{-1}\)Â K\(^{-1}\).
- 4.
The potential temperature of a parcel of fluid at pressure p is the temperature that the parcel would acquire if adiabatically brought to a standard reference pressure \(p_{0}\).
- 5.
In a Cartesian coordinate system fixed to the Earth’s surface, the vertical component of the Coriolis force is small compared to the gravitational pull. The former is therefore dropped in (1.10).
- 6.
Also referred to as the individual derivative.
- 7.
In the ocean, \(p_\mathrm{s} = p_\mathrm{a}\) at \(z=\eta \), where \(p_\mathrm{a}\) is the atmospheric pressure at the atmosphere/ocean interface. Setting \(p_\mathrm{a} = 0\) neglects the effect of the atmospheric pressure on the oceanic motion. It is important to include atmospheric pressure forcing when forecasting storm surges.
- 8.
\(R\ll 1\) is commonly referred to as the quasi-geostrophy condition [see (1.38)].
- 9.
The potential vorticity may also be derived in a similar fashion for a baroclinic fluid, but the mathematical expression is different (e.g., Griffies 2004, p. 70).
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Røed, L.P. (2019). Governing Equations and Approximations. In: Atmospheres and Oceans on Computers. Springer Textbooks in Earth Sciences, Geography and Environment. Springer, Cham. https://doi.org/10.1007/978-3-319-93864-6_1
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