Hydraulics and instabilities of quasigeostrophic zonal flows
Citable URI
http://hdl.handle.net/1912/5562DOI
10.1575/1912/5562Keyword
Shear flow; Fluid dynamics; ShearAbstract
The thesis addresses the applicability of traditional hydraulic theory to an unstable,
midlatitude 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 nonconstant, 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 subcritical 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
timedependent adjustment of an initially stable flow exhibits traditional hydraulic behavior such as control and influence in the farfield. 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.
Description
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution September 1994
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