On the Dividing Streamline Concept for Stratified Flow Past Three-Dimensional Axisymmetric Topography: Theoretical Advancements and Unsolved Questions

Abstract:

It is well known that, when a stably stratified flow with approach velocity U and buoyancy frequency N flows over an obstacle of height h, the low–level flow goes around the object while the rest flows over it for low F = U / N h. The streamline that separates the two types of flow is the dividing streamline, and the prediction of its height Hs is of great practical interest.

Sheppard (1956) provided the analytical solution Hs = h (1 – F) and, because of its practical utility, the formula continues to be largely employed, notwithstanding the criticism it has attracted because of certain underlying assumptions, viz., 1) the crude approximation of constant N and uniform approach velocity U, which is unrealistic for atmospheric flows; 2) the incorrect assumption of a complete balance between kinetic and potential energy at the mountain summit, which neglects the energy contributions of the perturbation pressure field as well as viscous dissipation adjacent to the hill surface. In this study, the first limitation is addressed by considering a logarithmic approach velocity profile but with constant N. A modified logarithmic velocity profile for stably stratified flows is proposed, and an analytical solution is obtained for Hs in terms of Lambert-W functions. Results are tested against smoke visualization experiments and related field measurements made during the Mountain Terrain Atmospheric Modeling and Observations (MATERHORN) Program. Some of the assumptions and perceived violations of them are tested using laboratory experiments conducted in a stratified water channel.