Baroclinic Instability and Transitions in a Two-Layer Quasigeostrophic Channel Model

Abstract: 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.