TABLE OF CONTENTS In this section we study the selective decay phenomena associated with the decaying barotropic quasi-geostrophic equations on the unit sphere (16.4) in the presence of Newtonian viscosity or hyper-viscosity or their combination, i.e. 
, the absence of external forcing ( 
? 0), and ground state topography (h ? W 1) . The reason that we can allow ground state topography is that a ground state topography merely changes the Coriolis force, so that it aligns with possibly another axis and with another magnitude (see Subsection 16.2.1). Thus all the identities that we need for the analysis and proof of selective decay remain unchanged, since the Coriolis forcing term is skew symmetric (see Chapter 3 and the argument below). We refrain from studying the case with general topography even though the general situation can be analyzed as well.
The result and approach here is very much similar to those in Chapter 3 for the flat geometry case. Thus we will omit most of the details and just state the result and sketch the main steps in the proof. A noticeable difference in the spherical geometry when compared with the flat geometry is that we are able to identify those initial data whose Dirichlet quotient approaches the first eigenvalue of the Laplace operator. This is again due to the independent dynamics of the ground state modes.
We follow the structure of Chapter 3 in this section. We first recall the basic dynamics equation.
