Barotropic instability at minimal resolution

Two linear minimum-resolution models of beta-plane channel flow are presented in analogy to the well-known two-layer model of baroclinic instability in order to see if basic features of barotropic instability can be demonstrated using similarly simple models. Two models are discussed within this pedagogical framework. A spectral model with two wave modes is applied to a cosine jet. Necessary conditions for instability are derived. The stability analysis shows that this simple model captures the shortwave cutoff and the asymmetry of the instability with respect to the direction of the jet quite well. It is demonstrated that the cutoff follows from Fjortoft's theorem for wave triads. A gridpoint model with two points in the interior of the channel is discussed as well. An analog to the classical necessary condition for instability is derived. A stability problem in a nonrotating system is discussed where the mean flow velocity is constant near the walls and a linear shear flow is assumed near the channel's axis. In this case, the stability characteristics of the low-order model come close to those of the full problem where a simple analytic solution is available. Addition of the 0 term stabilizes the flow. A proper choice of the initial conditions always enables short-term growth of the perturbation energy for stable mean flows. A qualitative interpretation of the instability mechanism is presented for both models, which exploits the fact that the locations of corresponding extrema of streamfunction and vorticity need not coincide. It is concluded that low-resolution models are well suited for a discussion of the basic features of barotropic instability.