TABLE OF CONTENTS It is an apparent fact in many instances, for example in the mesoscale and large-scale motions of the atmosphere and the ocean, that the fluid develops large-scale, coherent, and essentially two-dimensional flow patterns. These flows are the result of many competing effects, including dissipation and external forcing. Therefore, a fundamental problem is to understand how these large-scale flow structures develop through energy transfer to large scales for a fluid with dissipation and external forcing. In this chapter we study the existence and stability of large-scale flow structures for two-dimensional flows. In the simplest situation we assume that the flow is homogeneous (there is no density stratification) and it is described by the barotropic quasi-geostrophic equations with external forcing in the absence of topography and mean flow
We will assume that the stream function is doubly periodic, i.e.
and hence the velocity 
is doubly periodic as well. The case of channel geometry with periodicity in the longitude direction
and free-slip boundary condition (for the Newtonian viscosity case) at the latitude direction
can be treated in exactly the same fashion, where H denotes the height of the channel.
We will assume a fairly general dissipation operator, which includes Ekman drag, Newtonian viscosity, hyper-viscosity, or their combination
In the special case of having generalized Kolmogorov forcing
where ? is one of the eigenvalues of the Laplacian operator, the set 
in this case, and c.c represents the complex conjugate so that the forcing term could take the real...
