Hydraulics and instabilities of quasi-geostrophic zonal flows

Abstract

Abstract : The thesis addresses the applicability of traditional hydraulic theory to an unstable, mid-latitude jet where the only wave present is the Rossby wave modified by shear. While others (Armi 1989, Pratt 1989, Haynes et al.1993 and Woods 1993) have examined specific examples of shear flow hydraulics, my goal was to find general criteria for the types of flows that may exhibit hydraulic behavior. In addition, a goal was to determine whether a hydraulic mechanism could be important if smaller scale shear instabilities were present. A flow may exhibit hydraulic behavior if there is an alternate steady state with the same functional relationship between potential vorticity and streamfunction. Using theorems for uniqueness and existence of two point boundary value problems, a necessary condition for the existence of multiple states was established. Only certain flows with non-constant, negative dQ(%) have alternate states. Using a shooting method for a given transport and a given smooth relationship between potential vorticity and streamfunction, alternate states are found over a range of beta. Multiple solutions arise at a pitchfork bifurcation as a stability parameter is raised above the stability threshold determined by the necessary condition for instability. The center branch of the pitchfork is unstable to the gravest mode, while the two outer branches do not even have discrete modes. Other pitchfork bifurcations occur as higher meridional modes become unstable. Again, the inner branch is unstable to the next gravest mode, while the outer branches do not support this discrete mode. These results place the barotropic instability problem into a large set of nonlinear systems described by bifurcation theory.