In this article we present new experimental and theoretical results which were obtained
for the flow between two concentric cylinders, with the inner one rotating and in the
presence of an axial, stable density stratification. This system is characterized by two
control parameters: one destabilizing, the rotation rate of the inner cylinder; and the
other stabilizing, the stratification.
Two oscillatory linear stability analyses assuming axisymmetric flow conditions
are presented. First an eigenmode linear stability analysis is performed, using the
small-gap approximation. The solutions obtained give insight into the instability
mechanisms and indicate the existence of a confined internal gravity wave mode at
the onset of instability. In the second stability analysis, only diffusion is neglected,
predicting accurately the instability threshold as well as the critical pulsation for all
the stratifications used in the experiments.
Experiments show that the basic, purely azimuthal flow (circular Couette flow) is
destabilized through a supercritical Hopf bifurcation to an oscillatory flow of confined
internal gravity waves, in excellent agreement with the linear stability analysis. The
secondary bifurcation, which takes the system to a pattern of drifting non-axisymmetric
vortices, is a saddle-node bifurcation. The proposed bifurcation diagram shows a
global bifurcation, and explains the discrepancies between previous experimental and
numerical results. For slightly larger values of the rotation rate, weakly turbulent
spectra are obtained, indicating an early appearance of weak turbulence: stationary
structures and defects coexist. Moreover, in this regime, there is a large distribution
of structure sizes. Visualizations of the next regime exhibit constant-wavelength
structures and fluid exchange between neighbouring cells, similar to wavy vortices. Their
existence is explained by a simple energy argument.
The generalization of the bifurcation diagram to hydrodynamic systems with one
destabilizing and one stabilizing control parameter is discussed. A qualitative argument
is derived to discriminate between oscillatory and stationary onset of instability
in the general case.