An analytic solution for periodic thermally driven flows over an infinite slope: Defant (1949) revisited

The flow generated along an infinite slope in an unperturbed stably stratified atmosphere at rest by a time periodic surface temperature forcing is examined. Following Defant (1949), a set of equations is derived which extends Prandtl's (1942) theory to allow for nonstationary conditions. Uniform boundary conditions are conducive to an alongslope parallel flow, governed by a periodically reversing local imbalance between along-slope advection and slope-normal fluxes of momentum and heat. Solutions include both a transient part and a subsequent periodic regime. The former can only be expressed in an integral form, whereas the latter is a combination of exponential and sine or cosine functions of time and height normal to the slope. Key parameters are N_α = N sinα (where α is the slope angle, and N is the Brunt-Väisälä frequency of the unperturbed atmosphere) and the angular frequency of the driving surface temperature cycle, ω. Three different flow regimes may occur, namely subcritical (N_α<ω), critical (N_α= ω) and supercritical (N_α>ω). Properties of the solutions in each regime are shown.