Some Aspects of the Flow of Stratified Fluids: I. A Theoretical Investigation

The following paper is the first of a series of two relating to the problem of internal oscillations of a fluid in a gravity field with vertical gradients of density and velocity. The theoretical analysis in this paper will be supplemented by a report on an experimental investigation along the same lines to be published in a future issue of this journal. The exact, steady-state equations of motion and continuity of a perfect liquid moving two-dimensionally, with an arbitrary vertical distribution of density and velocity, are integrated once to yield a second-order differential equation. This equation is examined with regard to uniqueness and stability of the motion. A criterion is developed giving a sufficient condition for the motion to be uniquely determined by the configuration of the topography over which the fluid moves. It appears, further, that the condition of uniqueness is also a condition that a certain integrated quantity, called the kinetic potential of the motion, be a maximum. The suggestion is offered that this may correspond to a form of fluid instability. A detailed study is made in the special cases of a uniform basic velocity, and a certain type of shearing flow. In either case, it is shown that an internal Froudc number of about 1/3 divides the motion into two states, one of which is called supercritical, the other subcritical. From several viewpoints, these regimes are analogous to the corresponding states of flow of water in a channel. In the subcritical state the flow is in the form of standing wave patterns. When flowing supercritically, conditions may be favorable for the formation of internal “hydraulic jumps”. DOI: 10.1111/j.2153-3490.1953.tb01035.x