TABLE OF CONTENTS It is apparent that the earth is not flat, at least not on large scale. However, we have completely neglected the spherical geometry of the earth except the beta-plane approximation in the study of geophysical flows in the last 15 chapters. In this chapter, we will focus on geophysical flows on the sphere. In particular, we are interested in clarifying those properties that are parallel to the flat geometry case and those properties that are unique to the spherical geometry. The study of geophysical flows on the sphere is of great importance due to the geometry of the earth and other planets. We will see that most of the phenomena on the sphere have their counterparts for the flat geometry. However, there are certain features unique to the sphere due to the special geometry.
Unlike the periodic or channel (flat) geometry case, there are several useful and natural coordinate systems on the unit sphere S 2. We will utilize two coordinate systems in this chapter. The first coordinate system is an intrinsic coordinate system, the surface spherical coordinates
with 
being the longitude and ? being the latitude (see Figure 16.1). The second coordinate system is an extrinsic one with the latitudal variable ? replaced by z = sin ?
The two coordinate systems are related through the following relation
and we will work back and forth between the two coordinate systems depending on the convenience.
We first recall the...
