Dynamic Transitions of Quasi-geostrophic Channel Flow

The main aim of this paper is to study the dynamic transitions in flows described by the two-dimensional, barotropic vorticity equation in a periodic zonal channel. In [Z.-M. Chen et al., SIAM J. Appl. Math., 64 (2003), pp. 343--368], the existence of a Hopf bifurcation in this model as the Reynolds number crosses a critical value was proven. In this paper, we extend these results by addressing the stability problem of the bifurcated periodic solutions. Our main result is the explicit expression of a nondimensional parameter $\gamma$ which controls the transition behavior. We prove that depending on $\gamma$, the modeled flow exhibits either a continuous (Type I) or catastrophic (Type II) transition. Numerical evaluation of $\gamma$ for a physically realistic region of parameter space suggests that a catastrophic transition is preferred in this flow, which may lead to chaotic flow regimes.