Neutral and stratified turbulent boundary layer flow over low mountains

A uniform approximation of flow over gentle hills with a turbulent closure based on mixing length theory is derived. It permits to  describe the transition from neutral to stratified flow in the production of mountain drag. Our results corroborate previous studies showing that the transition from the form drag associated to the mountain induced changes in boundary layer friction  to the mountain gravity waves drag can be captured by theory. We also confirm that the first is associated with downstream sheltering with relative acceleration at the hill top, the second with upstream blocking with strong downslope winds.  We also show that the downslope winds penetrate well into the inner layer. The theory show that the altitude at which the incident flow need to be taken to calculate the drag is related to the inner layer depth at which dissipative effects equilibrate disturbance advection. We also show that the parameter that capture the transition, which in our case is a Richardson number, is directly related to the altitude of the turning levels of the gravity waves with respect to the mountain length. Our uniform solutions are also used to describe the wave field aloft and the distribution of the Reynolds stress in the vertical. Some directions to combine neutral and stratified effects in the parameterization of subgrid scale orographies in large scale models are given.