| Observations and numerical simulations both indicate that turbulent flows in the planetary boundary layer are often organized into coherent structures. These structures provide an efficient mechanism for mixing and transport of particulates, chemicals and energy in the form of heat and momentum. Consequently, an important problem in geophysical fluid dynamics is to understand the onset of turbulence from a given laminar state and the emergence of coherent structures from that process, and the structures that are most likely to emerge from sustained turbulent flow. In this work we investigate the emergence of coherent structures from stratified shear flows, such as are relevant to the atmosphere, using a generalized linear stability theory known as optimal perturbation analysis.; We develop the theory of optimal perturbation analysis, demonstrate the mathematical equivalence of different formulations, and consider the various formulations from the perspective of numerical implementation. Using these ideas, optimal perturbation analysis is applied to stratified shear flow. We consider idealized stratification and velocity profiles that are relevant to atmospheric dynamics. We show that the qualitative features of the optimal perturbation solutions are similar for different velocity profiles. The optimal perturbation structures are shown to become naturally two-dimensional, except at early optimizing times, and that they achieve significant energy amplification even in highly stratified flow. These results are in excellent agreement with observations of atmospheric turbulent events which appear to contradict the often-used Miles-Howard criterion, based on traditional asymptotic stability analysis. |
