  The bifurcation diagram of a truncation to six degrees of freedom of the
equations for quasi-geostrophic, baroclinic flow is investigated. Period
doubling cascades and Shil'nikov bifurcations lead to chaos in this model. The
low dimension of the chaotic attractor suggests the possibility to reduce the
model to three degrees of freedom. In a physically comprehensible limit of the
parameters this reduction is done explicitly. The bifurcation diagram of the
reduced model in this limit is compared to the diagram of the six degrees of
freedom model and agrees well. A numerical implementation of the graph
transform is used to approximate the three dimensional invariant manifold away
from the limit case. If the six dimensional model is reduced to a linearisation
of the invariant manifold about the Hadley state, the Lorenz-84 model is found.
Its parameters can then be calculated from the physical parameters of the
quasi-geostrophic model. Bifurcation diagrams at physical and traditional
parameter values are compared and routes to chaos in the Lorenz-84 model are
described.
