Timescales for Prandtl slope flow

Up-slope (anabatic) flows during the day and down-slope (katabatic) flows during the night are common features of boundary layers in mountainous/hilly terrain during fair weather conditions. The classical Prandtl slope flow model provides an analytical solution of the equations of motion and thermal energy for the simplest slope-flow scenario: flow of a uniformly heated or cooled boundary layer along a planar slope of unbounded extent. However, as the original Prandtl solution only pertains to the equilibrium (steady) state, it is not clear how long one would need to wait after the onset of the thermal forcing (e.g., after sunrise or after the sun disappears or reappears from behind a cloud deck) before it could be relevant. The present study exploits the exact solution of the unsteady version of the Prandtl model equations for a Prandtl number of 1 and the approximate solution for other Prandtl numbers to obtain the times for the wind and buoyancy fields to adjust to sudden changes in slope buoyancy. These response times are calculated from the unsteady solutions as the times to attain specified thresholds (percentages of each variable’s steady-state values) as a function of height, Prandtl number, and slope angle. The response times provide a basis for establishing whether, for a given slope angle, a quasi-steady state is even physically realizable within the daytime or nighttime phase of the diurnal cycle.