A Mathematical Analysis of Nocturnal Slope Flows
Oscillations are a persistent characteristic of katabatic winds or nocturnal slope flows, yet the physical mechanisms responsible for the oscillations are not yet fully understood. Numerical atmospheric models have also been found to produce oscillations in a slope flow setting. The goal of this study is to identify the basic physical mechanisms and corresponding parameter regimes that produce these oscillations. To accomplish the goal, a set of nonlinear ordinary differential equations (ODEs) that results from truncating the full governing partial differential equations is analyzed. Previous researchers hypothesized that the nonlinear ODE model supports the stable limit cycles that correspond to oscillatory behavior. The limit cycles were thought to arise from a Hopf bifurcation. This work shows that a Hopf bifurcation is impossible for the ODE model. Other researchers hypothesized that the fluctuations obtained from numerical models were due either to physical factors not included in the ODE model or possibly to the discretization. The latter possibility is plausible because the iterated map that corresponds to the commonly implemented numerical scheme for the problem is shown to support a Hopf bifurcation. However the present work shows that this bifurcation occurs only for time-steps outside the region of stability. Hence, any fluctuations in numerical results for a problem of this type would arise from numerical instabilities. A time-step algorithm, based upon the linearized problem, is developed to produce a time-step small enough to avoid numerical instabilities and hence the Hopf bifurcation.