An Instability of Shear Field-Buoyancy Field Interactions Leading to Vortex Formation

Abstract

With the object of gaining a better understanding of the mechanism involved in the formation of vortices in buoyancy-driven systems interacting with ambient vorticity, a mathematical approach based upon the hydrodynamical instability theory is employed. For simplicity, the mathematical model consists of an infinite horizontal line heat source with uniform ambient vorticity supplied. A governing stability equation analogous to the Orr-Sommerfeld equation is derived and solved by a Newton-Raphson iterative technique employing special matching conditions just outside the plume and symmetry conditions enabling analysis in a half plane. The solution predicts a standing-wave-type instability of the class suggestive of incipient vortices. Neutral stability curves are generated and a critical Reynolds number based on plume thickness is obtained for Prandtl numbers of 5/9 and 2. Based on neutrally stable disturbances, a representative perturbed flow streamline pattern is shown. The theory predicts a critical elevation above which unstable modes exist. An optimization analysis was performed in the domain of unstable disturbance modes which suggested an overall selection process. It was found that at one particular elevation in the plume, one particular mode of disturbance grows much more rapidly than all others at any location. This property is then related to the vortex periodicity in the observed phenomenon. An attempt is made to relate the analytical results to relevant field observations of various classes of geophysical vortices.