Local Barotropic Instability

Abstract

This paper investigates the modal and nonmodal instability of a barotropic jet streak. The normal mode analysis reveals that all the unstable modes are either stationary or propagating local modes. The more localized the jet is, the more dominant the stationary unstable mode would be. An exact analysis of the local energetics shows that the energy generation rate depends upon the local structure of the disturbance and the basic deformation field. The energy redistribution processes are the mechanical work done by the ageostrophic pressure and the energy advection by the basic flow. They affect not only the phase speed but also the growth rate of a normal mode disturbance by virtue of the zonal inhomogeneity of the basic flow. The local energy generation rate is maximum in the near exit region of the jet streak. The pressure work process contributes to an additional downstream shift of the maximum energy center and the advection process causes a further downstream displacement of the center. These three processes have comparable magnitude and tend to oppose one another locally. The compensating and yet accumulative effects of those three processes result in the downstream localization of an unstable disturbance. Our nonmodal analysis confirms that an isolated disturbance not only has to have a favorable orientation but also has to be in the downstream position with respect to the jet core before it can develop rapidly. Furthermore, a disturbance with a localized structure in the downstream region of the jet core can emerge from a zonally unbiased disturbance within a few days. The same mechanisms of local energetics account for the downstream localization of the disturbances during this transient adjustment as in the normal modes. The maximum instantaneous growth rate of such a nonmodal disturbance can be several times larger than that of the most unstable normal mode. The transitional stage can be understood in terms of simultaneous growth of and interference among the multiple unstable modes.