SHALLOW-WATER FLOW PAST ISOLATED TOPOGRAPHY .2. TRANSITION TO VORTEX SHEDDING

The formation of Karman vortex streets is studied within the framework of single-layer shallow-water dynamics and in absence of surface fric tion and background rotation. In the first part of this study, steady numerical solutions for flow past circular topography were obtained by imposing a symmetry condition that essentially suppressed vortex shed ding. In the second part, this symmetry condition is relaxed in order to study the transition into the vortex-shedding regime. This transiti on is due to an instability of the symmetric wake pattern. The most un stable global normal mode of this instability is derived by a numerica l method. Most of the features of this mode can be understood in terms of the absolute instability theory. The mode is essentially barotropi c and relies on a positive feedback between the perturbations located on the two shearlines on either side of the wake. The classical shear modes centered on a single shearline are, on the other hand, shown to be absolutely stable even though their convective growth rates are sub stantial. It is also shown that a recently suggested frequency selecti on criteria pertaining to absolute instabilities in slowly varying she ar flow successfully predicts the growth rate of the most unstable glo bal normal mode. Finite-difference numerical simulations are utilized to trace the evolution of the most unstable global normal mode. It is demonstrated that the evolution to finite amplitude of the most unstab le global normal mode leads to the breakup of the steady wake into an oscillating Karman vortex street. The frequency of eddy shedding in th e numerical simulations agrees well with that from observations of edd ies behind mountainous islands.