  The main objective of this article is to derive a mathematical theory
associated with the nonlinear stability and dynamic transitions of the basic
shear flows associated with baroclinic instability, which plays a fundamental
role in the dominant mechanism shaping the cyclones and anticyclones that
dominate weather in mid-latitudes, as well as the mesoscale eddies that play
various roles in oceanic dynamics and the transport of tracers. This article
provides a general transition and stability theory for the two-layer
quasi-geostrophic model originally derived by Pedlosky \cite{ped1970}. We show
that the instability and dynamic transition of the basic shear flow occur only
when the Froude number $F>(\gamma^2+1)/2$, where $\gamma$ is the weave number
of lowest zonal harmonic. In this case, we derive a precise critical shear
velocity $U_c$ and a dynamic transition number $b$, and we show that the
dynamic transition and associated flo0w patterns are dictated by the sign of
this transition number. In particular, we show that for $b>0$, the system
undergoes a continuous transition, leading to spatiotemporal flow oscillatory
patterns as the shear velocity $U$ crosses $U_c$; for $b<0$, the system
undergoes a jump transition, leading to drastic change and more complex
transition patterns in the far field, and to unstable period solutions for
$U<U_c$. Furthermore, we show that when the wavenumber of the zonal harmonic
$\gamma=1$, the transition number $b$ is always positive and the system always
undergoes a continuous dynamic transition leading spatiotemporal oscillations.
This suggests that a continuous transition to spatiotemporal patterns is
preferable for the shear flow associated with baroclinic instability.
