Hydraulics and instabilities of quasi-geostrophic zonal flows

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(ψ)/dψ 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. However, if the eastward transport across the channel is large enough, the normal modes may stabilize and these waves have a phase speed less than the minimum velocity of the flow. In this case, the flow is analogous to sub-critical hydraulic flow. The establishment of these states and the nature of transitions between them is studied in the context of an initial value problem, solved numerically, in which the zonally uniform jet is forced to adjust to the sudden appearance of an obstacle. The time-dependent adjustment of an initially stable flow exhibits traditional hydraulic behavior such as control and influence in the far-field. However, if the flow is unstable, the instability dominates the evolution. If the topographic slope renders the flow more unstable than the ambient flow, then the resulting adjustment can be understood as a local instability. The thesis has established a connection between hydraulic adjustment and the barotropic instability of the flow. Both types of dynamics arise from adjustments among multiple equilibria in an unforced, inviscid fluid.