Small-amplitude time-dependent motions of a uniformly rotating, density-stratified, Boussinesq non-dissipative fluid in a rigid container are examined for the case of the rotation axis parallel to gravity. We consider a variety of container shapes, along with arbitrary values for the (constant) Brunt-Väisälä and rotation frequencies. We demonstrate a number of properties of the eigenvalues and eigenfunctions of square-integrable oscillatory motions. Some of these properties hold generally, while others are shown for specific classes of containers (such as with symmetry about the container axis). A full solution is presented for the response of fluid in a cylindrical container to an arbitrary initial disturbance. Features of this solution which are different from the cases of no stratification or no rotation are emphasized. For the situation when Brunt-Väisälä and rotation frequencies are equal, characteristics of the oscillation frequencies and modal structures are found for containers of quite general shape. This situation illustrates, in particular, effects which are possible when rotation and stratification act together and which have been overlooked in previous investigations that assume that the vertical length scale is much smaller than the horizontal scales.

Small amplitude oscillations of a uniformly rotating, density stratified, Boussinesq, non-dissipative fluid are examined. A mathematical model is constructed to describe timedependent motions which are small deviations from an initial state that is motionless with respect to the rotating frame of reference. The basic stable density distribution is allowed to be an arbitrary prescribed function of the gravitational potential. The problem is considered for a wide class of gravitational fields. General properties of the eigenvalues and eigenfunctions of square integrable oscillations are demonstrated, and a bound is obtained for the magnitude of the frequencies. The modal solutions are classified as to type. The eigenfunctions for the pressure field are shown to satisfy a second-order partial differential equation of mixed type, and the equation is obtained for the critical surfaces which delineate the elliptic and hyperbolic regions. The nature of the problem is examined in detail for certain specific gravitational fields, e.g., a radially symmetric field. Where appropriate, results are compared with those of other investigations of waves in a rotating fluid of spherical configuration and the novel aspects of the present treatment are emphasized. Explicit modal solutions are obtained in the specific example of a fluid contained in a rigid cylinder, stratified in the presence of vertical gravity, with the buoyancy frequency N being an arbitrary prescribed function of depth.

The dynamical behavior of small-norm solutions to a simple fluid-loop model describing convection in a two-constituent fluid is discussed. Asymptotic techniques are developed, to describe the dynamic behavior of solutions near those parts of the linear stability boundary where one eigenvalue passes through zero (exchange of stabilities), where two eigenvalues become purely imaginary conjugates (Hopf bifurcation), and where two eigenvalues coalesce to zero.

We investigate the von Karman similarity equations for fluid flow between two infinite coaxial disks that rotate with equal rotation rates and in opposite directions. The nonlinear singular perturbation problem for high Reynolds number is analyzed by formal asymptotic methods. We construct an asymptotic solution valid away from the boundary layers that occur on each disk. This solution requires that the fluid away from the boundary layers is essentially nonrotating, and thus confirms a conjecture of Stewartson. Moreover, its properties agree precisely with estimates for a solution whose existence has been proven by McLeod and Parter.

We consider a simple fluid-loop model describing convection in a two-constituent fluid. The model permits explicit construction of linear stability and global stability boundaries in parameter space. A rigorous proof of global stability within the appropriate region is provided.

The time development of non-axisymmetric disturbances on a shear layer in a uniformly rotating fluid is studied theoretically. It is assumed that the Rossby number, Ekman number, shear-layer length scale, and initial disturbance amplitude are small, and that disturbances grow if vorticity transfer from the shear flow exceeds Ekman-layer vorticity dissipation, a mechanism investigated by Busse (1968). Expressions for the ultimate instability amplitude are determined for specified relations among the small parameters, using the volume-integrated energy equation, the shape assumption, and approximations for the spatial dependence of disturbances. Disturbance growth is limited by modification of the initially-unstable shear-layer profile, and the eventual amplitude is shown to be fairly insensitive to the specific form of the profile. Using data from observations by Hide & Titman, the predicted maximum velocity amplitude of the non-axisymmetric motions for their experiments is approximately one-quarter of the velocity of the shear flow at the point of maximum gradient.

The transient process by which an incompressible dissipative rotating stratified fluid adjusts to a small change in the rotation rate of its container is examined theoretically. The aim is to clarify the effects of the imposed density stratification and of the boundary condition specified for the density perturbation on the behaviour of the fluid, particularly during the time span when the adjustment is performed in a homogeneous fluid. For a weakly stratified fluid in a cylinder, it is shown how these two factors govern the nature and intensity of boundary layers on the vertical wall which close the secondary meridional circulation generated by Ekman layers along the horizontal boundaries. For a more strongly stratified fluid, the usefulness and importance of potential vorticity conservation in determining the quasi-steady motion is verified, and a calculation for a spherical container demonstrates some new features that arise only when the container boundaries are not normal or parallel to the rotation axis. It is shown that experimental results of Holton (1965) are in less good agreement with predictions of the linear theory than had been previously indicated.