Search Results

You are looking at 1 - 10 of 16 items for

  • Author or Editor: Robert R. Long x
  • Refine by Access: All Content x
Clear All Modify Search
Robert R. Long

Abstract

A discussion is given of a three-dimensional vortex in a viscous fluid. A theory is outlined that leads to a reduction of the steady-state Navier-Stokes equations of an incompressible fluid to three ordinary, nonlinear, differential equations. Although no numerical results are available, qualitative arguments based on these equations and on the form of their dependent and independent variables throw considerable light on the structure of the vortex. One important result is that the vertical velocity is of the same order of magnitude as the horizontal velocity.

Full access
Robert R. Long

Abstract

No Abstract Available

Full access
Robert R. Long

Abstract

The differential equations governing the unsteady motion of a system of two superimposed liquids are integrated by use of the method of characteristics. The solution differs significantly from that obtained by assuming a very thin lower layer.

If the basic shear across the interface is small, a wave breaks forward (as an ordinary water wave) only if the amplitude is fairly small and if the lower layer is thinner than the upper layer. If the upper fluid is thinner, the wave breaks backward. The speed of the various points of the wave does not vary monotonically with amplitude.

If a shear exists, higher velocities in the upper layer in the direction of wave propagation favor the forward breaking of a wave. The opposite shear may cause a wave to break backward.

Full access
Robert R. Long

Abstract

A theory is developed for the three-layer circulation in an overmixed estuary (finite fresh-water influx) or harbor (zero fresh-water influx) accompanying a two-layer structure in the large body of water outside. A determinate act of algebraic equations is derived for the general case and the form of the equations shows that for zero fresh-water influx the discharge q 1 from a harbor is proportional to the square root of the density difference between the two outside fluids.

The problem is solved completely when there is a uniform depth H of the fluids inside and outside the harbor, when the fresh-water influx is zero, and when the two layers of fluid outside the harbor are of equal thicknesses. The solution shows that the outflowing layer of water has a thickness d=H/2 and a flux q 1=HW(HΔb 0)1/2/8, where W is the width at the constriction and Δb 0 the buoyancy difference between the two outside layers of water.

A laboratory model reproduced the three-layer circulation of the theory. The outflowing fluid was quite turbulent and this made the observation of the layer thickness uncertain but it appeared to be close to the value d=H/2 of the theory.

Full access
Robert R. Long

Abstract

This paper presents a description of experiments on the stability of a polar vortex and offers a theory for the experimental results. The experimental set-up is a rotating spherical shell, containing two immiscible liquids. The denser liquid covering the polar region is given a rotation with respect to the spherical shell, and observations are made to determine the tendency for the vortex to be displaced equatorward. The observations indicate that a cyclone centered near the pole is stable in the sense that its center of mass remains near the pole. An anticyclone tends to be unstable, however, so that its center of mass is displaced equatorward if its rotation is sufficiently strong.

In connection with the experiments, the equations of motion of a simplified vortex were developed for a spherical earth. These are applied to obtain an instability criterion for a polar vortex. Other applications of these equations are described. One of these is a study of the motion of a vortex imbedded in a uniform vorticity field on earth. This reveals a tendency for all anticyclones and weak cyclones to move southward. This tendency is strongest at low latitudes.

The initial motion of a vortex in an environment at rest with respect to the earth is also considered. The net force on the vortex is very close to one-half the Coriolis force due to the internal spin.

Full access
Robert R. Long

Abstract

The experiment reported in this paper was designed to study the effect of large mountain-barriers on zonal currents in the atmosphere. Obstacles of various sizes and shapes are moved zonally in a rotating spherical shell of liquid, and observations are made of the resulting perturbed flow. Striking differences occur, depending on the direction of relative rotation of the barrier. If the flow is westerly with respect to the obstacle (in the direction of the basic rotation), a strong anticyclonic circulation develops around it and this sets up planetary waves around the globe. The wave number is found to correspond very closely to a frequency equation derived from the vorticity equation.

If the flow is easterly relative to the obstacle, a similar anticyclonic circulation arises, but is confined to the immediate vicinity of the barrier. No long waves occur but, if the motion is viewed with respect to the spherical bowls, the obstacle, moving eastward, drags a strong westerly jet with it in its latitude band.

A study was also made of the flow over a barrier which is not as high as the “atmosphere.” This reveals a strong anticyclonic turning of the fluid as it moves over the obstacle, and a tendency for this to combine with the anticyclonic circulation in the westerly case to produce a trough immediately downstream.

Full access
Robert R. Long

Abstract

This note relates the depth of the halocline in an estuary to the fresh water influx using simple and general arguments. It is shown that the depth becomes large for both weak and strong influxes. This result is similar to observations in a laboratory experiment and in the Alberni Inlet.

Full access
Robert R. Long

A description is given of the flow of a three-layer system of immiscible fluids over an obstacle immersed in the lowest liquid. A tentative suggestion is offered that some features of the motion may be similar to the flow of air over the Sierra range near Bishop, California. Some arguments are advanced supporting this suggestion.

Full access
Robert R. Long

Abstract

This paper contains a solution for a solitary disturbance in a westerly current on a “β-plane” earth. The current varies slightly with latitude and the existence of the phenomenon depends on this variation. The disturbance has a mathematical resemblance to the solitary water wave of Scott Russell. It is a single ridge if the current increases with latitude, and a single trough if the current decreases with latitude. The speed of propagation depends on the amplitude.

Full access
Robert R. Long

Abstract

An investigation by G. I. Taylor, of the steady motion of an obstacle along the axis of a rotating fluid, is extended in this paper. It is shown that Taylor's particular solution is just one of an infinity of functions comprising the general solution.

The theory is applied to motions in a rotating cylinder of fluid. A critical Rossby number is derived, below which the flow around the obstacle is wave-like. When the Rossby number is greater than the critical value, the flow consists only of a local perturbation that dies out rapidly on both sides of the obstacle. Various other critical numbers exist, below which additional modes of oscillation become dynamically possible.

An experiment was designed to test the theoretical results of this paper. An obstacle was moved along the axis of a long cylinder of rotating water. The resulting flow patterns were observed visually and photographically. The three-dimensional wave motions which occurred in the experiment were unquestionably the same as those in the theoretical solution.

Full access