The Earth's climate is a remarkably complex physical system; constructing models to study it is a difficult task which requires parameterization of a multitude of physical processes. Not surprisingly, such models quickly become difficult to understand due to the vast number of nonlinear processes that are active in them.
Therefore, an important line of work in atmospheric science involves the development and use of intelligently chosen idealized models, designed to better understand the results of comprehensive climate models as well as the fundamental dynamics of atmospheric circulations. These models are simpler to interpret than the full climate models, but hopefully can still provide insight into the dynamics of their more complex cousins.
In this talk, we give a summary of some topical problems in climate dynamics, and the hierarchical modeling approach we have used to study them. We will discuss physical problems such as the predicted poleward shift of the midlatitude jet stream with global warming, and changes in energy fluxes and temperature gradients in the atmosphere. Focusing on the effect of moist convection on these issues, we present a variety of idealized models that we have used to study these problems. These range from models of 3-D fluid motion on a rotating sphere in the presence of condensation, to highly idealized 1-D PDE models of diffusive energy transport.