Some Aspects of the Flow of Stratified Fluids: III. Continuous Density Gradients

The investigations of Parts I and II are extended to include experiments and theoretical considerations relating to the behavior of fluid systems with continuous gradients of density. The general equation of steady-state motion derived in Part I is integrated to yield the flow of a stratified fluid over an obstacle of finite dimensions. The results indicate a more or less complicated laminar wave motion for obstacles of maximum height below a certain value. Larger barriers cause an overdevelopment of the waves to a point where closed circulations and negative horizontal velocities appear. It is shown that this is accompanied by locally unstable distributions of density and eventual turbulence. If the height of the barrier is further increased, the velocity increases indefinitely in some parts of the field, becoming infinite for an obstacle of a certain size. No steady-state solution exists for larger barriers. The two critical values of the obstacle height depend primarily on the Froude number: If this number exceeds 1/π the solution exists and is stable for all obstacles; for small Froude numbers the barrier must be small if the solution is to exist or be stable.

If the obstacle is small enough to permit laminar or moderately turbulent motion, the accompanying experiments verify all important features of the theory with remarkable fidelity. Larger obstacles cause considerable turbulence and blocking effects which propagate upstream, causing alternate maxima (jets) and minima of horizontal velocity in the vertical.