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”.

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.

A description is given of the flow of two superimposed layers of fluid over a barrier. This represents a partial experimental investigation of a problem considered theoretically in Part I. In general three regimes of motion are possible: If the velocities of the fluids are sufficiently small the interface is little disturbed except for a slight depression over the barrier. If the velocities are sufficiently high the interface swells symmetrically over the obstacle. At intermediate speeds a hydraulic jump occurs in the lee of the barrier and the lower layer increases in depth upstream. Two occurrences do not fit into the above description: If the obstacle is small compared to the depth of the lower layer, weak lee waves appear at low speeds, increasing in amplitude as the approach velocity of the fluid is increased. This seems to be the only case in which perturbation theory provides an adequate prediction of the flow. The second anomalous occurrence is the appearance of a “jump down” or hydraulic “drop” in the lee when the speed of the fluid is moderately high, the obstacle large, and the upper fluid relatively thin. The description of the experiments is supplemented by a theoretical discussion, employing the assumption of a hydrostatic pressure distribution. In general this theory provides a satisfactory explanation of the observed behavior. The paper concludes with a discussion of meteorological implications.

The following paper is the first of a series of t w o relating to the problem of ititcrnal oscillations of a fluid in a gravity field with vertical gradients of density and velocity. The theoretical analysis in this paper will be supplenicnted by a report o n an experimental investigation alons the same lines to be published in a future issuc of this journal. T h e exact, steady-state equations of motion and continuity of a perfect liquid nioving twodimensionally, with an arbitrary vertical distribution of density and velocity, arc integrated once to yield a second-order diffcrcntial cquation. This equation is examined with regard t o uniqueness and stability of the motion. A criterion is developed giving a sufficient condition for the motion to be uniquely dcterinined 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 thc motion, be a niaxiniuni. T h e suggestion is offered that this may correspond to a forni of fluid instability. A detailed study is made in the special cases of 3 uniform basic velocity, and a certaiu type of shearing flow. In either case, it is shown that an internal Froude number of about 1/3 divides the motion into t w o states, one of which is called supercritical, the other subcritical. From several viewpoints, these regimes are analogous to the curresponding states of flow of water in a channel. In the subcritical state the flow is in the form of standing wave patterns. W h e n flowing supercritically, conditions may be favorable for thc formation of intrrnal "hydraulic jumps".

A solution is given for a viscous vortex in an infinite liquid. Similarity arguments lead to a reduction of the equations of motion to a set of ordinary differential equations. These are integrated numerically. A uniform feature is the constant circulation K outside the vortex core, which is also a viscous boundary layer. The circulation decreases monotonically towards the axis. The axial velocity profiles and the radial velocity profiles have several characteristic shapes, depending on the value of the non-dimensional momentum transfer M. The solution has a singular point on the axis of the vortex. The radius of the core increases linearly with distance along the axis from the singularity, and, at a given distance, is proportional to the coefficient of viscosity and inversely proportional to K.Finally, a discussion is given to indicate that intense vortices above a plate, like the confined experimental vortex, or above the ground, like the atmospheric tornado and dust whirl, will not resemble the theoretical vortex except, possibly, far above the plate.