# Dynamical foundations

2.1 Dynamical and thermodynamic laws

Since the atmospheric and oceanic circulations take place within a rather thin layer of fluid on the surface of the Earth, large-scale motions in the atmosphere and oceans can most conveniently be described in spherical coordinates. The basic equations have been discussed in many standard textbooks; we will assume that the reader is familiar with them and therefore these equations will be introduced here in a concise way. One of the best sources is the book by Holton (2004), An Introduction to Dynamic Meteorology.

2.1.1 Basic equations

The momentum equations

In a rotating frame of reference (Fig. 2.1), Newton's second law can be written in vector notation as

Dt p

where u = (ui, vj, wk) is the three-dimensional velocity vector; Du /Dt is the total derivative or the "material derivative" of velocity, i.e., the rate of time change for an observer who rides with the fluid particles; ¿2 = a>j (j is a unit vector) is the vector representing the rotation of the Earth; p and p are pressure and density; g is the gravity vector; and F is the friction force, such as the surface wind stress, bottom drag and internal friction.

The convention of spherical coordinates used in physical oceanography is (X, 0, r), where 0 < X < 2n is longitude, — n/2 < 0 < n/2 is latitude, and r is radius. In spherical coordinates, the total derivative of velocity is