Abstract
The formation of Kármán vortex streets is studied within the framework of single-layer shallow-water dynamics and in absence of surface friction 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 shedding. In the second part, this symmetry condition is relaxed in order to study the transition into the vortex-shedding regime.
This transition is due to an instability of the symmetric wake pattern. The most unstable global normal mode of this instability is derived by a numerical method. Most of the features of this mode can be understood in terms of the absolute instability theory. The mode is essentially barotropic 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 substantial. It is also shown that a recently suggested frequency selection criteria pertaining to absolute instabilities in slowly varying shear flow successfully predicts the growth rate of the most unstable global 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 unstable global normal mode leads to the breakup of the steady wake into an oscillating Kármán vortex street. The frequency of eddy shedding in the numerical simulations agrees well with that from observations of eddies behind mountainous islands.
